A fluid dispenser pump comprising: a pump body; at least a first piston that slides in leaktight manner in said pump body on each actuation so as to expel a dose of fluid through a dispenser orifice (45) that is provided in a dispenser head (40); a closure member (38), movable between a closed position and an open position, being disposed in said dispenser head (40) upstream from said dispenser orifice (45), said closure member (38) being urged towards its closed position by a resilient element, such as a spring; said closure member (38) in the closed position co-operating in leaktight manner with said dispenser head (40) in at least three distinct sealing zones (381, 382, 383).