As panache45 and others note, if it indeed is the case that there is a fixed finite number of people, each of whom kills precisely one person, with no one killed by more than one person, then it must indeed be the case that everyone is killed. If this is the stipulation, then this is what happens.
As to the question of who kills the last person in such a scenario, it must either be that someone manages to kill after they themselves are killed (e.g., by planting a time-bomb), or that several people manage to kill simultaneous with their own death (e.g., a Mexican standoff with no survivors). The latter might well be seen as a special case of the former, with slight jitter of timing (e.g., a bullet released from one’s gun might kill others after one’s own death).
If we outlaw the possibility of simultaneous or posthumous killings, then… it cannot actually be the case that everyone kills one other person (with a fixed finite number of people, with no one killed by more than one person). Them’s the breaks.
[We can discuss other scenarios, of course, but they won’t actually involve everyone killing one other person (many scenarios under discussion, for example, involve some people killing before they manage to kill anyone). Once we allow scenarios where not everyone kills one other person, there is lots of room for considering different such scenarios with different traits, so we should not expect any One True Answer among them.]