You've started this thread in RPG General. What system(s) are you playing?
I have quite a bit of one-on-one combat in my games. Sometimes the PC wins; sometimes s/he loses. Typically the stakes aren't death, and so losing doesn't mean that a PC is dead.
Here are two links to Classic Traveller episodes that involved 1:1 combat. In the first, the PC in question ended up feeling "badass" because he got to escape capture (after wrestling with his captor to take control of the latter's gun) and gain control of some powered armour. In the second the PC in question was humiliated by a childhood acquaintance.
And here are two links to Prince Valiant episodes that involved 1:1 combat. In the first, two PCs were bested in jousts by the best knight in Britain; the third PC won a freak victory (mechanically, by spending a player auto-win resource). In the second, a PC fought a joust to prove the truth of an accusation against a NPC, and lost. So the PCs had to concede that the NPC was innocent, the result being that the NPC was able to kill his brother and take control of Fort Seahawk.
I don't have a link to a write-up, but I remember a Burning Wheel session we played which also involved a duel to prove the truth of an accusation. In that case the PC lost too.
Of the above episodes, only the first involved a real risk of death, although Classic Traveller combat does not have to be deadly. Maybe you want to consider how you're establishing stakes in your game?
