(assuming United States law here, though I'd be surprised if it were significantly different in other jurisdictions with such restrictions)
Your friend is incorrect: that would be a new offense, for which Person A could be prosecuted anew. If your friend's logic were correct, once a person is convicted of robbing a store, they'd be free to rob that store without repercussions for life.
It's worth noting that the conviction isn't relevant: the prohibition of double jeopardy in the United States preventprevents even multiple prosecutions (except, in some cases, for separate state and federal prosecutions or foreign prosecutions).