In mitosis, the telomeres shortern and eventually the hayflick limit takes over and the cell dies - does anything like this occur during binary fission?

Answer 1

Additionally, the Hayflick Limit is a phenomenon involving the telomeres on the ends of human linear chromosomes. Telomeres are present to buffer against natural shortening that occurs during replication of the genome. Specifically, this is a function of the linear nature of the human chromosome.

Moreover prokaryotes have circularized chromosomes. They haven't any need for buffering against the shortening of their genome. If such shortening were to occur, then the impact would be immediate as it would excise coding regions of their genomes.

Also, it is worth noting that replication of the genome is separate from reproduction. Human somatic cells may replicate their genomes during the S Phase of a typical cell cycle. However, they are not then undergoing reproduction, sexual or otherwise. Whereas binary fission is a means of asexual reproduction found in Prokaryotic species.

Answer 2

Binary fission is an asexual reproduction method amongst prokaryotic cells, a more primitive form of cell. The typical bacterium will live about 12 hours, dividing an average of five times per hour. So while there is a time limit so to speak there is nothing that directly corresponds to the Hayflick limit.