What we know about cosmic eschatology makes true immortality seem unlikely, but there's plenty of time (as it were) to develop new theories, make new discoveries, or find possible new solutions. See:

Cirkovic "Forecast for the Next Eon: Applied Cosmology and the Long-Term Fate of Intelligent Beings"

Adams "Long-term astrophysical processes"

for an excellent overview of the current best-estimate for how long a human-complexity mind might hope to survive.

Just about everything CIrkovic writes on the subject is really engaging.