"No, the woman says."
Me too I wonder, because in English, you could say "the woman says" or "says the woman". Both work.
I think the "no" should have been in quotes as this is reported speech : '"no", the woman says'. Italian sometimes moves the subject around to stress it. In this case, the speaker wants to stress that the woman said "no" and a way to do it is to move the subject from it's natural place.