I would say as a standard answer that yes it is possible for a mentally ill person to be a doctor.

However you seem to have an analytical mind. Have you given any thought as to how you would handle the grueling hours and stress for almost the decade of schooling? What about financial issues? Would you need to work while in medical school?

What is your motivation to become a doctor? If it’s for admiration and kudos you are setting yourself up for failure. If you have a deep and profound need to help people there are many other options that could be rewarding for you.

All that being said, if you are determined to do this I say go for it.