To make Mud go away in Gothic 1 Remake, just whack him a few times with your fists. The guards and other NPCs won’t care, and after a couple of hits, he’ll run away and stop following you.
Get Rid of Mud: NPC Removal details

How To Make Mud Go Away
Equip Your Fists
Pull out your little fists from your weapon menu. You don’t need any weapon—just bare hands work perfectly.

Hit Mud Multiple Times
Whack Mud a few times (2–3 strikes).

A couple of hits is enough—he doesn’t need to die. After hitting him, he’ll turn and run away on his own.

Once you hit him a couple times, Mud should go away and stop following you. The nearby NPCs don’t react strongly to this, so no bigger scene occurs.

Critical Mistakes to Avoid
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Don’t hit him too early: Exhaust all Mud’s dialogue options first before beating him. If you beat him too early, he stops talking entirely and you lock out the “Mud Freak Achievement”.
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Don’t rely on one hit: A single attack may not always work—you need a few punches to stop him from trailing you permanently.
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Watch for witnesses: If you kill him, ensure no guards or NPCs are nearby to witness your actions.

Mud Freak Achievement Tip
To unlock the Mud Freak achievement, listen to all 24 unique dialogue interruptions from Mud. Several lines activate only after interacting with specific NPCs or choosing certain dialogue options. Exhaust every conversation option before taking action against him.
What Happens Later? SPOILERS!
Mud is destined to die no matter what you do. He gets tragically murdered by another NPC in Chapter 4. Even if you beat him up early, the story plays out normally and he dies in Chapter 4 anyway. Most players won’t even make it to Chapter 4 with Mud still alive since monsters often kill him while exploring.
Getting rid of Mud is simple: just punch him a few times in camp and he’ll disappear. This lets you explore in peace without his nonstop chatter ruining your adventure.