On our cruise in October this question came up. The ship does not stamp passports but they said you can go to the post office in the town and get it stamped. People were doing it but they spent most of their time in port chasing around just to get a stamp.
The more places you take your passport the more chance you have to lose it.
You can try two things. One is asking Guest Relations; some may say no, some may say yes.
The other is to go _not_ to the Post Office but to the Immigration office in dock (ask around if it's not evident right away.) We've gotten our US passport stamped in Grand Cayman, Ocho Rios, Nassau, and Torotla that way.