A subassembly for a dental care or surgical handpiece comprises a push button (16) comprising a contact surface (23) adapted to receive a manual pressure able to actuate the push button (16) slidably in a sliding direction. This subassembly further comprises a piece (21) for retaining the push button (16), as well as a bayonet device (28a, 33, 35) able to keep assembled the push button (16) and the retaining piece (21).