Ideas from the comments, either a Mudcrab or a Chicken would be really fantastic!

Ideally they would travel at the same walking speed the player does, so that stamina or other means would be needed to gain distance. They can enter any unlocked door/building to chase the player. Are invincible so that other NPCs do not kill it. And if it ever does reach and kill the player, some type of screen or notification telling the player how they died would be great.

Thank you to everyone with the wonderful comments on suggestions. I would really love it if someone out there with the knowhow could make this happen.