I can give you one well known example of this. Alexander the Great almost died in the Battle of Granicus (334 BC) but one of his officers - Cleitus the Black - saved his life. Fast forward to 328 BC when Alexander was at a drinking party. Now, accounts differ based on which one you're reading but most agree on how things went down from here. Alexander's generals were discussing how Alexander was better than certain gods, praising him, etc. Cleitus the Black intervened, saying that this comparison is blasphemous; the Greeks found it extremely problematic whenever a Greek raised himself above the level of another Greek, especially so when he compared himself to a god. He told Alexander that his generals were just lying to him to get his favor. Alexander killed Cleitus there and then; although he immediately regretted this, he couldn't change what had happened. So yes, this relatively minor disagreement ended in Alexander the Great killing a man who saved his life.

Source: "The Greeks: History, Culture, and Society" by Ian Morris

I'll just point out that this isn't limited to the Ancient World. People kill each other over "not something to kill someone over" all the time, though. There's an example of this exact same thing (two people not giving way on a single track road) from 19th century Corsica in Roger V. Gould's Collision of Wills (I highly recommend that book--it's sociology, but uses a fair bit of history, anthropology, and criminology all mixed together). The sociologist and criminologist Jack Katz wrote an excellent book called The Seductions of Crime about crime and emotions. The first chapter, called "Righteous Slaughter", is full of examples of similar instances of murder from the modern U.S.--people killing each other over parking spots and the like. I strongly, strongly recommend the first two chapters of that book (the other chapters are good, but for the most part not as great).