The invention relates to a pump comprising a piston (1) which slides in a body (3) having a chamber (5) defined between an inlet valve (11) and an outlet valve (12), comprising an outlet valve element sliding during actuation in a sealed manner in the chamber (5), which comprises passage means (40) such that, at the end of the actuation, said outlet valve element (39) cooperates in a non-sealing manner with said passage means (40) in order to open said outlet valve (12), said inlet valve (11) comprising an inlet valve element (10) which slides after closure of the inlet valve (11) in a sleeve (9) of the body (3) having a reduced diameter and containing a spring (20) bearing on said inlet valve element (10) and on a bottom of said sleeve (9), said spring (20) returning the piston (1) into its rest position, said sleeve (9) having an inner diameter of less than 4.2 mm, said spring (20) having a force of at least 20 N, so that said pump delivers said fluid product at a pressure (P) of at least 15 bars.