What's Mud's deal?
Mud is an Old Camp digger in Gothic 1 Remake who follows you around and won't stop talking to you. He's not a bug. He's not broken. He's doing exactly what the developers intended, and he's been doing it since the original Gothic in 2001. The remake keeps him and, if anything, turns the volume up.
Don't kill him. He looks like a nuisance, but he has real content later.
Why he follows you
Mud is lonely. He's the lowest-ranking digger in the Old Camp, and you're the first person who didn't tell him to get lost within the first ten seconds. That's the whole reason. He latches on and follows you around the camp, talking at you whether you respond or not.
There's no dialogue option that permanently stops this. You can walk away, you can ignore him, you can leave the camp. He'll be there when you come back. The remake's NPC system makes this more noticeable because Mud's dialogue can overlap with other conversations.
If you attack him, he'll flee rather than fight — he's a digger, not a guard. The game has a knockout system where NPCs go unconscious and get back up later, and defeated NPCs can shift to a hostile or angry attitude afterward. In the original Gothic 1, punching Mud was the classic way to get him to stop following you: he'd get up angry and keep his distance. The remake preserves the knockout system, so this likely still works, but his exact post-defeat behavior may differ from the original. Either way, knocking him out won't kill him or lock you out of his content — only finishing him off while he's down does that.
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Mud fights in the Old Camp arena against Gor Hanis and Kharim. He's not a strong fighter — he's a digger with a digger's gear — but the fights exist and they're part of the arena content. If you're exploring the Old Camp arena system, Mud is one of the participants you'll see.
He matters later
Don't kill him. Mud has a scripted role in the late game that you won't see coming from his early-chapter behavior. Saying more would spoil it. Just know that the annoying digger who won't leave you alone eventually earns his keep.
He's killable — the game lets you — but doing it removes his arena content and his late-game participation. The annoyance is temporary. The content is permanent.
If you genuinely can't stand him following you, leave the Old Camp. He doesn't follow you outside the camp walls. He'll be at his stool near the arena when you come back, ready to talk your ear off again.
For where to go and what to do in the Old Camp, see the Old Camp guide. For who to join and why, see which faction to pick.