IELTS without passport?

Hi

I have been living in the UK for 3 years and I find it difficult to renew my visa recently. I made an application which was unsuccessful so I am officially overstaying now.
I heard about the Canada immigration program and I would like to apply. I was wondering if my overstaying status can affect my application with the Canadian program.

Before all, I have to apply for an ielts test for the language purposes. Can I sit the exam without a valid visa in the UK? Rules changes everyday with the Home Office and I DON'T want to be embarrassed while writing the exam.

Thanks for replying