You do
not need your passport for the medical exam. My husband didn't have a passport when he did his exam (this was in India). Actually, on the advise of our MP's office, we submitted the PR application before my husband even had his passport since he had already been waiting more than 1 year for it. You will need some type of identity document though. This is a quote from the medical exam instructions of what you need to bring with you:
Passport and photocopies of the biographical data pages (this is the page that shows the date of birth, the country of origin, etc.)If no passport is available, provide an official identity document bearing a photo and the date of birth of the family member