It is typically believed that Wobbuffet live in caves to hide their tails, a source of embarrassment, from other pokemon and trainers. In reality, the reverse is true: Wobbuffet live in caves in the faint hope of hiding other pokemon from their tails.
Wobbuffet were initially a truly pacifist species, but this left them little more than punching bags, so in the rough and tumble world of pokemon they were forced to develop Counter and Mirror Coat for self-defense. Indeed, even these techniques Wobbuffet use only reluctantly, and they feel guilt whenever they knock out another pokemon. But their tails, symbiotic spirits which absorb a Wobbuffet's pain, burned for revenge on all living things. It is these tails which developed Shadow Tag, forcing pokemon into battle with a Wobbuffet only wants peace. It is these tails which learned Destiny Bond, slaying those unfortunate pokemon slower than Wobbuffet who still manage to slay one. And it was these tails which caused Wobbuffet to retreat into darkness, hiding from foes and hoping passersby can sneak past their tails unseen.
Most pokemon trapped by a rebellious body part would rip their tails off and endure the wound, but reclaim their freedom. Dugtrio and Magneton, after all, live in harmony, and it is sadly common to see a one-headed Dodrio walking around after a three-headed fight to the death. Wobbuffet possess this power as well, although they must use a great deal of psychic power to rip their tails off, for their arms are incapable of performing this task. They refuse for the same reason they refuse to attack with techniques such as Psychic: because they can not live with hurting anyone else save for in self-defense, even if it would save many others.
Even if it would save themselves.