Overnight train from Basel Switz to Florence or Rome? Wondering if there is such a thing and could I use an ItaliaRail ticket or would I need another ticket.....

Hello, thank you for your question. 

There are overnight trains from Switzerland to Italy, currently Basel is not a destination that has been released to us yet on our site at italiarail.com41, we do have Zurich but, if you are looking for Basel you may wish to try the trainline's website at www.trenitalia.com35 as they should have the destination available for booking.