Cut the Safety Rope

Alice is dangling over a drop large enough to be certain death. The only thing holding her up is Bob, who she's attached to by the safety rope. Despite his assurances that he'll save her, she knows that the weight of the two of them is too much; she's as good as dead and if she doesn't do something, Bob will die too.

So ignoring his words, she takes out her knife, says a final goodbye and cuts her safety rope.

Although the safety rope version is the most common, this trope covers any situation where someone is in such deadly danger that their death is all but assured and any rescue attempt would more than likely result in more death, making the mercy killing of the victim the pragmatic solution. A More Hero Than Thou dispute may have preceded it, or the victim acted quickly to pre-empt the question.

Usually it's the victim who makes a Heroic Sacrifice by deliberately condemning themselves to save others from death or prevent them from attempting a risky rescue attempt. The would-be rescuers are likely to feel a great deal of angst and guilt even if they did everything they could to save them.

The alternative is that its the potential rescuer who condemns the victim to their fate. While potentially just as pragmatic as if the victim made the call, this will usually paint the character in a very negative light, more so if they actively cause the victim's death; less so if they do so at the victim's request.

As a Death Trope, Spoilers ahead may be unmarked. Beware.

Examples of Cut the Safety Rope include:

Anime and Manga

Grenadier: Yajiro's love interest deliberately refuses to Take My Hand so as not to pull him off when she's blown off a cliff in a flashback.
Although it's not a literal rope, this is basically how Transformers Armada ends, in a rare case of it happening between enemies: Optimus Prime and Galvatron are about to be devoured by Unicron, and Optimus is the one clinging to a ledge with one and and holding Galvatron up with the other. Galvatron stabs Optimus' hand, forcing it to let go and plunging himself into Unicron; then, evidently, without Galvatron around, Unicron doesn't have enough evil to feed off of, so he disappears as well. Yes, Galvatron saved the universe.
A variation almost happens toward the end of Fullmetal Alchemist, when Wrath and Greed/Ling are both hanging off the edge of the Briggs fortress. Ling's devoted vassal Lan Fan is holding onto her lord, whom Wrath is trying to pull down with him. Lan Fan recently had an automail arm attached to her body and the weight is tearing it off, so Ling tries to order her to let go, but she refuses. Luckily, a Briggs soldier shoots Wrath and causes him to fall alone, enabling them to pull Ling back up before Lan Fan loses her arm entirely.
Subverted in Detective Conan. In the mermaid case, Kazuha falls over a cliff, but Heiji jumps after her, grabs her with his one hand while holding onto a overhanging branch with his other. Kazuha, realizing the branch won't hold them both, takes an arrow she received as a prize recently and stabs Heiji's hand with it, in order to make him let go. He holds on nonetheless.

Comic Books

Tintin in Tibet: Captain Haddock attempts this, but drops his pocket knife. Tharkey, who has walked out, catches up to them and makes his presence known, before saving them off-page.

Film

A variation occurs in Cloudy with a Chance of Meatballs: In order to shut down the FLDSMDFR, the heroes have to go down a long pit lined with shards of peanut brittle. Since Sam is allergic to peanuts, Flint has to go alone, and he uses a licorice rope to jimmy down, with Sam feeding the rope as he goes. In the process, she gets stabbed with one of the shards, and starts to have an allergic reaction. She refuses to leave Flint, even though she'll die without her medicine, so Flint bites through the licorice and falls the rest of the way down, after telling Brent to make sure Sam gets back to the plane.
Vertical Limit: Happens to the main characters father in the prologue, causing the friction between him and his sister. All the more heartbreaking as it was the son who cut the rope at his fathers insistence just moments before the sister was able to attach additional securing that might have saved them. Repeated with an Ironic Echo at the end of the film when The Mentor sacrifices both himself and the Smug Snake (unwillingly) to save the others.
The Day After Tomorrow: One of the rescue party falls through the glass roof of a snowed over in mall they were inadvertently hiking over. Cuts his rope to save the others when he sees the rest of the glass will crack before they can pull him up.
Touching The Void had more or less the opposite. When trying to return to base because one of the pair had a broken leg, the other was forced to cut the rope and let them fall while attempting to lower them. Amazingly, despite thinking that his partner had certainly died from the fall, he not only survived (with even more injuries) but managed to crawl back to camp.
- Its a true story, and even more amazingly the man who did the falling (Joe Simpson) has since publicly supported the actions of the man who did the cutting (Simon Yates) and forgiven him. Simon Yates is yet to forgive himself.
In the Disney action-adventure Iron Will Will and his father have a dog-sledding accident on the bank of a river. With the two of them and all the dogs tangled in the sled/ropes and being dragged into the river, and Will screaming for him not to do it, the father cuts himself off the sled and drowns while the dogs drag Will and the lightened sled to safety.
Mission to Mars: One character takes off his helmet in space to keep his wife from wasting her fuel to come and save him as she would not have had enough to get them back.
Played straight pretty much to the letter in Alien Resurrection. Clone Ripley and her new True Companions made up of space-smugglers are escaping from a Xenomorph in an unused elevator shaft. One smuggler, Vriess, is paralysed from the waist down so his crewmate Christie strapped him to his back to get through a flooded section without leaving him behind. During the ladder climb, Christie makes the mistake of looking down... just in time for the alien to spit acid into his face. He reflexively lets go of the ladder to claw at his melting face but Vriess catches the ladder, revealing that his paralysis gifted him with higher than average upper-body strength. Jorner eventually kills the alien with Guns Akimbo but the critter grabs onto Christie's foot, weighing down on Vriess whose arms can't support their collective weight. Christie, despite the protests of Vriess, cuts the straps binding them together and plummets to the bottom of the shaft where he's instantly killed on impact.
- Somewhat justified: the acid blinded Christie so he would've just slowed the group down. Letting him stay behind would've resulted in being captured and implanted.
The Guardian featured this in its climax. After Ben (played by Kevin Costner) rescues Jake (played by Ashton Kutcher) on the ship in the middle of the storm, they take the safety rope up to the helicopter together. But the rope is not designed for their combined weight and begins to break. Ben decides to sacrifice himself to save Jake. However, instead of cutting his part of the rope, he simply unstraps the glove that Jake is holding onto, falling into the water from a fatal height. Though, if the end of the movie it to be believed, Ben may be Not Quite Dead, simply Ascending to A Higher Plane of Existence.
In The Mountain, Spencer Tracy as a famous mountaineering guide refuses to lead a rescue up a mountain to a crashed plane because it's simply too dangerous at that time of year. Another guide, and friend of his, leads them because they simply can't leave the survivors up there. Part of the rescue party comes back and tells them it happened just as Tracy warned them and that his friend had sacrificed himself to save the others.
In Volcano, two of men who are rigging a condo to blow and create a lava barrier from rubble get trapped in the parking garage underneath it. When they get the radio call that the building has to be blown up right now, they lie and claim that they've gotten clear so the explosives can be set off in time.

Live-Action TV

Ghost Whisperer had a variation in one episode, where a father allowed his wife to die (at her explicit request) in order to save their daughter. A pair of malicious manipulative ghosts who had hung around the house where they had killed someone else and then each other, were able to use the guilt of both father and daughter, to oppress and live through them.
This is used in The Reveal of an episode of MacGyver: a rock star is revealed to be her twin sister impersonating her after the latter died during a climbing accident and she cut the rope. This caused the surviving twin to have a Freak-Out and start switching between the two identities to "keep her sister alive".
The Doctor does a variation of this in the Doctor Who episode "The Satan Pit" not to save someone's life, but to satisfy his curiosity. Fortunately, while he almost certainly fell far enough that he would have damaged or killed himself, some Benevolent Precursors gave him a way to cushion his fall and allow him to save the day.
- Played straight by one of the Thals in The Daleks when he realizes Ian isn't strong enough to save him.
An episode of M*A*S*H had BJ forced to do this as the rescuer. Granted, the situation was the chopper pilot telling him he had to, they were already overloaded, and the thing they needed to rescue the wounded soldier from was a group of North Korean troops. Much angst and guilt.
In Power Rangers Lightspeed Rescue, we have Ryan and his father holding onto something that can't support both of them. Captain Mitchell lets go. (Ryan manages to save him, though.)
In Stargate Atlantis, Sheppard won't leave behind a soldier who has been partially fed on by a Wraith and is in no condition to run or fight. In the end, the soldier shoots himself so Sheppard won't die protecting him.
- The same thing happened with McCay and another scientist, with whom he constantly argued. The other scientist was partially fed on by an ancient Wraith, so McCay stays behind to guard him, while Sheppard goes out to hunt for the Wraith. Realizing Sheppard needs McCay's help and that, at best, he's looking at a short life in bed, the other scientist shoots himself, allowing McCay to go help Sheppard. While McCay doesn't do much more than distract the Wraith for a few seconds, this proves to be enough for Ford to do a Gunship Rescue and blast the Wraith with a drone weapon.
This is how Juliet from Lost dies. She falls over the edge, only to be grabbed by her boyfriend Sawyer. Unfortunately, Sawyer begins to slip, so she lets go to save him.

Video Games

In the videogame Prince of Persia: The Sands of Time, Farah is hanging over a cliff by the handle of a dagger, and the Prince is holding the blade, cutting his hand with enemies approaching—but she knows that this will cause both of them to get killed, so she lets go and falls to her death. Fortunately the dagger is able to rewind time...unfortunately it's out of sand.

Web Comics

Three Word Phrase: As a result of a Minor Injury Overreaction.

Western Animation

Inverted in Avatar: The Last Airbender, when Sokka and Zuko break into Boiling Rock. The warden of the prison is shown early on pointedly stating that he would rather jump into the boiling lake just outside the prison than allow his escape-free reputation to be tarnished. At the climax of the episode, the good guys take the warden hostage so they can take a gondola over the aforementioned boiling lake to safety unharassed. However, the warden manages to free himself enough so that he can yell at the guards to cut the lines holding up the gondola, knowing full well that he'll be boiled alive along with the would-be escapees when it falls. The attempt is foiled, but points for trying.