A fluid dispenser member (2), such as a pump, including a body (21) that is provided with an inlet sleeve (22) that defines an axis X, a dip tube (3) having an end (31) that is fitted in the inlet sleeve (22), and a decorative unit (7) including a decorative element (72) that extends in the proximity of the dip tube (3). The unit also includes a support member (71) that is fastened around the inlet sleeve (22), the support member (71) including a reception housing (711) for receiving the inlet sleeve (22) and localized connection means (712) that are situated beside the reception housing, the decorative element (72) being connected to the support member (71) at the connection means (712).