"Infinity" poses a ton of problems for both science and philosophy, I'm sure,
One might mean either of two things by "infinitely divisible time." One might mean merely that (1) any nonzero interval of time can in principle be divided into smaller and smaller units indefinitely: what's sometimes called a "potentially infinite" collection of units of time each of which has nonzero duration. Or one might mean that (2) any nonzero interval of time actually consists of infinitely many -- indeed, continuum many -- instants of time each of which has literally zero duration: what's sometimes called an "actually infinite" collection of instants. I myself favor (2), and I see no good reason not to favor (2) over (1).
Both views of time are controversial among philosophers, and some physicists conjecture that both views are false (they conjecture that an indivisible but nonzero unit of time exists: the "chronon"). But let's apply (2) to the time of a person's death. Classical logic implies that if anyone goes from being alive to no longer being alive, then there's either (L) a last time at which the person is alive or a (F) first time at which the person is no longer alive. If (2) is true, then there can't be both L and F, because according to (2) no two instants of time are adjacent to each other. In other words, if L exists, then there's no earliest instant at which the person is no longer alive; and if F exists, then there's no latest instant at which the person is still alive. According to (2), there are instants other than L that are arbitrarily close to L but no instants right next to L. Ditto for F.
To put it another way: (2) implies that no transition is literally from one instant to the next, because there's no such thing as the next instant. This includes the transition from being alive to no longer being alive. Nevertheless, exactly one of L or F exists. Which one is it? I don't know the answer to that question, but I'm not sure it's a well-posed question anyway.