Assuming that the prohibition is directed at one individual visitor, B is right. The absence of indications of temporary violation(s) being allowed defeats A's presumption that the prohibition refers only to the count of visits as of certain time/date.
For a person to visit the place thrice, that person necessarily has to violate the prohibition. Simply put, there is no such thing as a person's double-visit by which he can transition from having visited the place once to having visited it thrice.
By contrast, it is entirely feasible for a person to have one dollar and be paid two dollars with a two-dollar bill, whence he goes from having one dollar to three dollars without it being possible to identify an instant where he only had two dollars. Unlike the impossibility of a person's double-visit, there are two-dollar bills. Therefore, the counter-example formulated by @user4894 is mistaken.
In the dialogue of that counter-example ("Do you have two dollars? No, you have three dollars"), that response "No, you have three dollars" is actually answering the different question of "How much [or how many dollars] do you have?". But when a person has three dollars and is asked "do you have one/two/three dollars?", the correct answer in each case is yes. Similarly, when a person has visited the same place thrice and is asked "did you visit the place once/twice/thrice?", the correct answer is yes. And the affirmative in the questions with "twice" and "thrice" reflects that the prohibition has been violated.
The actual ambiguity in the OP's question is only whether:
B's answer reflects a conjecture that the prohibition is by person (not by place), whereas A's answer does not help to solve that ambiguity. Regardless, whatever conjecture(s) A and B may have are external and therefore irrelevant to what the actual question: the answer follows and never determines the question.