A fluid dispenser comprising: a fluid reservoir (1) provided with an opening (11); a fluid dispenser member (2), such as a pump or a valve, said member comprising a body (21) and an actuator rod (23) that is displaceable down and up along an axis X; a fastener ring (3) for fastening the dispenser member on the opening of the reservoir; a pusher (4; 4′; 4″) that is axially displaceable down and up so as to displace the actuator rod (23); and a dispenser endpiece (50) defining a dispenser orifice (51); displacement means (6, 4; 6′, 7; 6″, 7, 4″) that are suitable for displacing the endpiece (50) both in turning about the axis X and in radial translation, the radial distance between the endpiece and the axis X varying as the endpiece turns about said axis X, such that the endpiece is displaceable between an extended position remote from the axis X and a retracted position close to the axis X, the displacement means comprising a rotary actuator member (4; 7) that turns about the axis X, the endpiece (50) being constrained to turn with said actuator member, and a cam path (62) that is prevented from turning relative to the reservoir (1), the radial distance between the path (62) and the axis X varying along the path, the endpiece (50) being engaged with the cam path in such a manner that it follows the cam path while it is being turned by the actuator member (4; 7), the fluid dispenser being characterized in that the cam path (62) is formed by a cam element (6″) that is mounted on a ferrule (8) that is mounted in stationary manner relative to the reservoir (1), the cam element (6″) being prevented from turning relative to the ferrule (8), but being capable of being displaced axially, the pusher (4″) being in engagement with the ferrule (8) via a second threaded cam path (84), the pusher (4″) being turned by the actuator member (7), the pusher (4″) being displaced axially relative to the ferrule (8) and to the actuator member (7) by following the second threaded cam path (84) while said pusher is being turned by the actuator member, such that turning the actuator member simultaneously causes the rotary axial displacement of the pusher and the rotary radial displacement of the endpiece.