I have never played in a game where PCs were really out to kill each other, trading blows back & forth, or waging a multi-session campaign against each other.
I have engaged in a little bit of serious character conflict, however, though it never turned violent. The climactic session in the last campaign I played featured a cleric of the Broken God (Domain of the maimed, deformed, disfigured, diseased, deranged) who
claimed the mantle of the Broken God after freeing him from his bondage. I, playing as wizard of the good persuasion, attempted to prevent this particular vacancy in the pantheon from being filled. I fictionally wasn't positioned to do anything about it apart from casting a spell, though, so I had to succeed on my Perfect Summons but I failed the roll. Once the cleric had achieved Godhood, I did not have any moves left to me but Ritual, which I triggered. GM ruled I needed the tears of the God of Healing to fix my deranged cleric-cum-god friend, so I Perfect Summons-ed the God of Healing to beseech him for his tears. Ultimately, the GM ruled the God of Healing wasn't going to accede to my request so I had my wizard dedicate himself to the service of the God of Healing to thwart the effect of the new Broken God on the world, and chose the new class of cleric.
We, of course, retired those characters at that point.
All this is to say the GM principle "Be a fan of the characters" can be tricky to fulfill when the characters are pulling the GM in different directions. I myself would hope never to GM a true arena-style PVP session.