In my opinion, if the many-worlds interpretation of quantum mechanics is correct, we will certainly experience inmortality in some way. But what I imagine is very different from what you are picturing. Branching of universes doesn't have to happen in dichotomic life/death situations, it would be happening in every moment, thousands of millions of times per second, as all the fundamental processes happening everywhere are quantum.

At this moment, some quantum process in my neurones has decided whether I was typing this sentence now or a milisecond later. Two universes have been created, one in which I typed it at 18h24m05,000s and another one in which I did it at 18h24m05,001s. None of the copies of myself will ever communicate between each other so all of them feel they are still living in the only possible universe.

Of course this take us to endless possible universes, and in an infinity of them I will live forever. You are saying our body is finite but that doesn't have anything to do with it. For example, to not get old, it just takes to stay in the branches in which the irreversible biological processes that make us decay, which are quantum at their microscopic nature, don't happen. There is a ridiculously small but non-zero chance that that happens, so the branching will occur, and it could keep occuring forever with millions of variations within it. In some universe I will be 1000 years old and have a Coke and in another one I'll be having a Pepsi.

Am I my future copies? I wouldn't say so, but all of them are me. The copy of me whose quantum working brain decided not to write this comment ten minutes ago is not me anymore, but we both are the "me" that existed ten minutes ago. The copies of you people in that other universe won't have to go through this boring comment.