A fluid dispenser member, including a chamber and an outlet valve having an actuator rod that is axially movable down and up, and on which a sleeve slides, under the effect of a pre-compression spring and the pressure of the fluid, over a maximum axial stroke. The effective axial stroke of the sleeve is dependent on the force of the spring and the pressure of the fluid in the chamber. The actuator rod includes side outlet mechanism that is suitable for being closed and uncovered selectively by the sleeve, the side outlet mechanism extending over an axial height that is greater than their radial width.