As ditching the group is not an option, here are a few ideas about how to solve this problem that you're describing. I hope that any of this will be for any help…
Change the type of the campaign
A nice solution, especially if it doesn't seem like they're ever gonna stop with this is to simply change the campaign. There are rule-systems and games that are tailored to deal with this kind of behavior. Some of them promote killing each other (like the great Paranoia), while others just change the rules and arenas in which it will take place (Vampire: the Requiem can deal with player conflict quite great, especially since most of it is deal below the radar or one comes into problems). Other possibilities include gladiatorial campaigns where the PCs will have to fight each other for their survival. For an extra dose of irony, go for Best Friends.
If the players ever get bore of destroying the lives of each other, those rule-systems also tailor other types of campaigns, which are far less character-conflict centric (yeah, even Paranoia, if you go for the extra-straight type of game). So this can be a nice bridge to bring them back to more normal games. Vampire is especially good for this, as it is a mainstream game, which promotes under-radar activities but doesn't require it and can be quite good for both conflict heavy and group-centric kinds of play.
Talk with your players
My go-to solution is to have a conversation with the players. Talk with them, understand why they're acting this way or the other and explain to them what your problem with the situation is. It may seem quite silly, but it is one of the best ways to change the characters'/GM's perspective about topics and it is great to illustrate and through this to solve the problems that bug the participants' minds.
Present an enemy who is willing to take the credit
As I understand, each one of them wants to be the killer. This is great for you, because that by presenting an enemy who is far more powerful than each one of them alone, he can use their conflicts to steal the credit for killing characters. Make sure that your players understand that without joining forces he will be the one responsible for their killing, that he is going to steal those kills from them, and there's a huge chance that they're going to combine forces in order to kill him/her before this enemy kills them and steal the kills.
It is also a common trope of many great stories; the people who can't stand to be near each other have to combine forces in order not to perish. It will give this story a more epic feel and if you will present it rightly, they will feel like they're part of a movie.
It all serves a dark purpose
Another option is to give them some hints, which grow bigger and bigger, that this intra-party fighting is serving the purpose of some villains. Then make the players hate those villains (by kill stealing, for example) and you've got something in your hands. By making them understand that this fight helps those they hate, there's again a huge chance that they're going to stop fighting each other and start to focus on other matters, like killing those kill-stealers.
Prove to them that other types of play are fun also
Make a deal with them, "you will try for a session or two not to kill each other, not to fight each other, and to try to enjoy some of the things that I'm gonna present to you. I'm sure that you're going to enjoy some of those things even more, like murder mysteries, or interrogations or conversations with NPCs or being a part of a huge story or even a romantic subplot. If you won't like it, I won't push again, but if you will do, you will surely thank me later…" Then prove to them that it is more fun. Give them one head-blowing kind of a session and they will look at those times of fighting with disdain, feeling glad that it is beyond them.