A rabbit that eats paint?!?

My gf has a rabbit that is kept in a closet space. The problem is that the rabbit seems to love gnawing on the paint. It does have pieces of wood to chew on but the paint must seem irresistable to the rabbit (named Jack Bauer btw). So, 1)just how bad do you think the paint could be to the rabbit? and 2) anyone... show more My gf has a rabbit that is kept in a closet space. The problem is that the rabbit seems to love gnawing on the paint. It does have pieces of wood to chew on but the paint must seem irresistable to the rabbit (named Jack Bauer btw).
So, 1)just how bad do you think the paint could be to the rabbit?
and 2) anyone have any creative solutions??
Update: The rabbit gets plenty of playtime outside the cage. It shouldnt even be called a cage because the closet space does not have a door. We put a baby gate in front of the closet so it doesnt get out. It has its own food, water, stuff to chew on etc....but still chooses to eat paint!
Update 2: Of course I realize paint is bad for you or anyone for that matter...what I asked was: HOW BAD is it for the rabbit!
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