Do immortal organisms exist?
I now found this Wikipedia article on biological immortality. It's pretty much what I was looking for.

Wikipedia describes the phenomenon as follows:


  Biological immortality refers to a stable or decreasing rate of
  mortality from cellular senescence as a function of chronological age.
  Various unicellular and multicellular species may achieve this state
  either throughout their existence or after living long enough. A
  biologically immortal living thing can still die from means other than
  senescence, such as through injury or disease.


It has a list of such organisms as well, consisting of


tardigrades
hydras
turritopsis nutricula, some sort of jellyfish
lobsters* 
planarian flatworms


Addendum: This blog post takes a good look into the myth of lobster immortality. It seems to be as much a myth as the result of any substantial observation.