Chin's and paint

I'll probably sound like a terrible owner in the next paragraph or so, I know.

I have two chinchillas and they love to come out for play time, but one of them is starting to concern me with paint. He has started chewing the paint off my shelf and baseboards. I was told by a chinchilla owner at my local pet store that this was completely fine as long as it was nontoxic paint, but I just can't imagine that being true. Wouldn't it get stuck in their intestines and clog it and stuff, just like plastic? I try my best to stop him every time, but I'm still worried this is dangerous for him despite what the people at the petstore said. Do I need to cover my baseboards? I can definitely do that. I'm not sure how to cover my shelves, but I'll find a way if it is actually neccesary. I really have no clue whether it's a real problem or not, which I know I probably should. I just trusted my local pet store owner and didn't cover the baseboards before I got the chinchilla. Please respond ASAP, I wanna know whether I need to cover it as soon as possible. (The paint is definitely nontoxic, by the way, but I really don't know if that actually makes a difference).