Non-Permissive Use Car insurance - will it cover my accident?

What state are you in ? I've never heard of "non permissive use". Normally, someone with a valid license and permission to drive an insured car is covered under that cars policy. I guess you will have to read the policy. If you can't see where it says "non permissive use" then I would ask the insurance company to point it out to you,. If they won't or can't and still will not pay, and contact your insurance commissioner/and or get a lawyer