I don't know of physics rules ruling this out. However, I suspect this doesn't resolve the problems that the people I know who care most about immortality are worried about. (I'm not sure – I haven't heard them express clear preferences about what exactly they prefer on the billions/trillions year timescale. But they seem more concerned running out of ability to have new experiences than not-wanting-to-die-in-particular.)

My impression is many of the people who care about this sort of thing also tend to think that if you have multiple instances of the exact same thing, it just counts as a single instance. (Or, something more complicated about many worlds and increasing your measure)