The Help Center article on answer bans doesn't say anything like that. (This is where the ban message directs you to look.)
The MSE FAQ says you can (but I doubt that anyone but "power users" reads this):
If you're unable to improve your existing posts, you'll get the chance to post one new question or answer 6 months after your last question/answer.
I'm not banned but I am confused. Can we edit one of these to make it clear how the system works? (Once we figure out how the system works.)
In contrast, the Help Center article on question bans says this:
If you're unable to improve your existing questions, you'll get the chance to ask a new one 6 months after your last question. If that question is positively received, you may be able to continue asking questions; if not, then the ban will be reinstated.
If this is how answer bans work too, then we can reuse this verbiage (with obvious edits) in the other article.