The person is a Director of Company A. On the side they also do work for Companies B and C (unrelated to Company A).The answer to this is yes you can pay a director for self employed services, if they are genuinely self employed and provide services to other businesses in addition to yours (as a minimum).
However, this is usually very rare circumstances and would need a thorough review of what exactly the director is being paid for.
The person is a Director of Company A. On the side they also do work for Companies B and C (unrelated to Company A).
Are you saying that this person can be paid as a self-employed for Company A (which I believe is what the OP is asking) or that they are an employee of A (and must be paid PAYE) but can work for/be paid by B & C as self-employed?
If the work you refer to is as a Director then the answer is no, it isn't possible. As a Director of your company doing work as a Director they are an employee and would have to be PAYE.I want to make a family member a director of my company. However, I would like him to work and pay him as self employed. Is this possible?