Maybe I'm naive, but I would politely mention it to her first. If she gets snippy and doesn't stop, then I would bring it up to the professor. I was a college prof for many years, and my reaction would be if someone complained to me is "Have you mentioned it to her?" Poor thing. She sounds like she could use a visit to a counselor for some anxiety issues. That said, you shouldn't have to put up with her distraction.

Do people really hurt people these days over being told that their gum-chewing is distracting? Good grief! Methinks the mental illness problems in this world are definitely escalating!