A fluid dispenser device for mounting on a reservoir. The device has a dispenser member with a body and an actuator rod that is axially movable down and up with the body; a fastener ring for fastening the body on an opening of a reservoir; and a dispenser head mounted on the actuator rod. The dispenser head includes a connection sleeve engaged axially on the actuator rod, the head including a radial duct, an orifice formed in an outer casing that surrounds the connection sleeve, defining a substantially annular gap. The fastener ring includes an axial guide for preventing the dispenser head from turning freely, while enabling it to move axially. The dispenser head includes a guide element that co-operates with the guide, the guide element extending radially from the connection sleeve to the outer casing, in the annular the gap.