--May I use your pen?
A. with pleasure B. of course
The answer given is B.
But I think A also works here.
What do you think?
But here is an entry from Oxford Dictionary:
“May I sit here?” “Yes, with pleasure.”
For the question, you don't think WITH PLEASURE is suitable; is it because the question starts with MAY but not WOUlD?
Had the question been "Would/Could/Will you lend me your pen?" then "Yes, with pleasure" would be a perfectly good answer.
May I use your pen?
Yes, of course [you may].
Could you lend me your pen?
Yes, [I will lend you my pen] with pleasure.