A fluid dispenser comprising: a fluid reservoir (1) provided with an opening (10); and a dispenser member (2, 3, 4, 5, 6), such as a pump or a valve, mounted on the opening (10) of the reservoir, the member forming a fluid chamber (C) that is provided with an inlet valve (28, 43), an outlet valve (47, 511), and a lip (42) that is movable in leaktight sliding in a cylinder (27) between a rest position and a depressed position, so as to cause the volume of the chamber (C) to vary, and the fluid in the chamber to be driven through the outlet valve towards a dispenser orifice (50); said container being characterized in that, in the proximity of the depressed position, the inlet valve defines a first escape path (F1) that puts the reservoir (1) into communication with the chamber (C), the chamber, in the proximity of the depressed position, communicating with the outside through a second escape path (F2) that passes via the outlet valve and the dispenser orifice so that, in the proximity of the depressed position, the reservoir (1) and the chamber communicate with the outside through the open first and second escape paths (F1 and F2), and with the outlet valve closed.