Someone who persists in the face of any and all possible setbacks (short of the cessation of his life) is a plain "madman". Or a "robot", or a "machine".
(Someone who persists in the face of any and all possible setbacks *including* his death is a fictional figure such as a zombie or ghost.)
I think the other answers deal with normal individuals who do have some upper bound on their perseverance, and therefore do not keep going despite setbacks, only despite some setbacks.
Of course, everyone keeps going despite some setbacks.
Those who are fighters, steadfast, and so on (any admirable names for those who display courage, persistence and fortitude), keep going despite setbacks that would stop most people, but do not keep going in the face of all possible setbacks.
Someone who goes irrationally in the face of setbacks is a "fool", or "gambler". (Perhaps literally a gambler: a gambler regards losses to be temporary setbacks on his or her way to regaining a lucky streak, and keeps on gambling.)
Someone who keeps going in the feeding of a self-destructive habit, in spite of severe setbacks, is an "addict". The slang term "[X] junkie" is someone addicted to [X], whatever [X] is.
If it is not so self-destructive, perhaps "maniac" or "bum". For instance a "ski bum" keeps going to the slopes in spite of setbacks like not being able to maintain a regular job. "X bum" for any X means someone who persists in doing only X, and is not detracted by taking showers, grooming, or holding down a job.
Someone who persistently persists in some beliefs in spite of setbacks such as the consequences that come with alienating other people, is a "lunatic", "crackpot" or "X freak" for some X, for instance "religious freak".
Someone who chronically persists in living off others in spite of setbacks is a "parasite" or "moocher".
Someone who persists in annoying others in spite of setbacks, such as having numerous online accounts closed, or being evicted from apartment to apartment, is a "sociopath". Online he or she may be a "troll" or "spammer".
All of these people regard themselves as overcoming "challenges of life". The addict's challenges of life are where the next "hit" comes from. The gambler's challenges of life are swings between losses and wins, and the ensuing emotions, and so forth.