A fluid pump includes an input shaft, a pumping unit comprising a pump rotor, and a clutch arranged between the input shaft and the pump rotor. The clutch comprises at least two input clutch disks, at least two output clutch disks, a permanent magnet element, and an actuator. The at least two input clutch disks and the at least two output clutch discs together define at least two clutch liquid gaps which are filled with a magneto-rheological clutch liquid. The permanent magnet element shifts between an engaged position wherein a magnetic field of the permanent magnet element penetrates the at least two clutch liquid gaps with a high magnetic flux, and a disengaged position wherein the magnetic field of the permanent magnet element is less than in the engaged position. The actuator moves the permanent magnet element between the engaged position and the disengaged position.