There are many organisms that are technically biologically immortal. However, I would like to point out that the definition of biological immortality is this:
...cells that are not limited by the Hayflick limit, where cells no
longer divide because of DNA damage or shortened telomeres.
(That's from here.)
So biological immortality doesn't really cover disease or physical trauma, which would actually include damage to DNA. I know what you're thinking: But wait, doesn't biological immortality literally mean cells that don't die because of DNA damage? Nope. Close, but no. Look back up at the definition. You probably caught it this time (by the way, it's not listing two different reasons (DNA damage and shortened telomeres), just rewording the Hayflick limit (DNA damage aka. shortened telomeres)). Biological immortality does not mean that the cell will not die because of excessive mutation (which would fall under physical trauma mind you, be it from chemicals, radiation, or just plain mistakes during replication), it means that the cell will not die from excessive programmed mutations, aka. its Hayflick limit, aka. telomere degradation. A molecular biology course summary from Berkeley says:
However cells that express telomerase still undergo cellular senescence in response to DNA damage, oncogenes, etc.
Ok, sidetrack time (go ahead and skip this part if you already know what telomeres are). A telomere is a repeated DNA sequence at the end of a chromosome that protects the coding regions from deletion (the reason for these deletions is complex and off topic, but long story short, RNA primers cannot attach to the very end of a chromosome, so a little of the lagging strand is lost during every replication). So the telomere takes the hit instead of the important coding DNA further up the chromosome (see this article.) As you can imagine, there is only enough telomere to go around, so cellular aging exists unless the organism has active telomerase, which is an enzyme that adds telomeres every time they are lost (see this article).
So, short answer, biological immortality is very possible in its actual definition (see above).
Concerning your definition (immortality regarding all mutations, programmed or otherwise), an organism comes to mind: Physarum polycephalum. It is biologically immortal in the way we have just defined, but also avoids DNA mutation by sharing DNA (constantly comparing and fixing sequences) among thousands to millions of nuclei through homologous recombination, which repairs DNA mutations at a freakishly high efficiency when you include millions of strands bearing what should be the same sequence (see this article). Physarum polycephalum is a slime mold, meaning that when cells meet each other, they fuse. This particular slime mold actually prefers to be a plasmodium, which is a constantly growing mass that is technically a multinucleated single cell but can grow to be potentially infinitely large as far as we can tell. Because of its extreme ability to correct mutations, this organism has been referred to as, exactly like you said, an "evolutionary dead-end." I'm sure it still sustains some mutations over a very long period of time, but you might want to check it out nevertheless.