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"Invictus" by William Ernest Henley embodies this trope. He wrote it when he was 14 years old, and he was getting a leg amputed because he had contracted tuberculosis in the bone. He almost had the other one amputed as well, but he protested and got another doctor to save it after lengthy surgery over that leg's foot. He didn't lose his spirit, though, and he kept a strong, jovial and masterful attitude throughout his life; so much so, that family-friend Robert Louis Stevenson was so impressed with his maimed yet commanding figure that he served as inspiration for Long John Silver.
Rudyard Kipling's poem "The Overland Mail" portrays a postman in India as this, in terms that make "neither rain nor snow nor glo m of ni t" seem kind of mild (bear in mind the poem specifically states he's doing all this at night ... in the jungle ... uphill):
Is the torrent in spate? He must ford it or swim. Has the rain wrecked the road? He must climb by the cliff. Does the tempest cry halt? What are tempests to him? The service admits not a "but" or an "if." While the breath's in his mouth, he must bear without fail, In the Name of the Empress, the Overland Mail.
Also, in his poem, "If-"
If you can make one heap of all your winnings And risk it on one turn of pitch-and-toss, And lose, and start again at your beginnings And never breathe a word about your loss: If you can force your heart and nerve and sinew To serve your turn long after they are gone, And so hold on when there is nothing in you Except the Will which says to them: "Hold on!"
"Epitaph on an Army of Mercenaries" by A. E. Housman invokes this, referring to the British Army in World War I. (Apparently the German press had been saying that the British soldiers, professionals rather than conscripts, should be considered mercenaries.) "What God abandoned, these defended" — you can't get much more Determinator than that.
As the page quote for the main page suggests, Ulysses (or Odysseus, if you prefer) is definitely one, whether in Alfred Lord Tennyson's Ulysses or in Homer's The Iliad and The Odyssey.