Beginning with 2 assumptions:
1. The universe is infinite - either a single universe (a) temporally or (b) spatially infinite or an (c) infinitude of temporally and spatially disconnected universes.
2. The universe follows a distribution of matter (or the state of that matter) that is cosmologically analogous to a normal sequence of digits
Normal Number

Is this sufficient to establish an instance of subjective human self-awareness, identity, as immortal?

Identity, here, consists not of a single string of connected moments in time (a branch), but of a tree of these branches that all rise from a common root history.

Some portion of my experienced history may die out (states with limited or non-existent physically possible branches), but those that do not will perceive themselves as me. Example - I am in a plane that fails catastrophically at 30,000 ft mid-Pacific. Those branches after the catastrophic failure are almost all ruled out. My continued identity will persist with, among others, those branches where the plane did not fail catastrophically.

Immortality would be ruled out only if all possible branches from the tree root are eventually pruned, such as if I was child who realized self-awareness only after the catastrophic failure of the airplane.

This is not quite the same as quantum immortality, but may suffer from some or all of the same concerns. This premise does not rely on a many worlds interpretation, but instead on a 'resonance' across spatially and temporally disconnected spaces. This resonance is a product of, given the 2 assumptions, all possible states being realized an infinite number of times.