Lady Gaga -- Do What U Want With My Booty

The "no touching" rule did NOT apply for this lap dance -- Lady Gaga shoved her ass in her boyfriend Taylor Kinney's face Friday in NYC ... and his hands were EVERYWHERE.

Gaga -- wearing fishnets and a thong bodysuit -- went full stripper on the "Chicago Fire" star inside 42West's new nightclub BPM. We're told Kinney even spanked her.

A source in the VIP area tells TMZ, Gaga showed up with Kinney just after midnight, and the two were all over each other pretty much the whole time.

Don't stop. Let's party.