A progressing cavity fluid motor for driving a drill bit for deep drilling tools is described. Such a motor is mounted on a drill string and is powered by a fluid such as drilling mud. The pump includes a housing, a stator with female helical threads within the housing and a rotor with male helical threads mounted inside of the stator. The drill bit is connected to the rotor. In the present invention the rotor has a threaded surface which is formed of an elastically deformable sleeve supported by a carrier shaft. The sleeve is mounted on the carrier shaft in such a manner as to prevent rotation between the two so that the sleeve drives the shaft by positive engagement between these two elements. Means are arranged for introducing a pressure inside of the elastically deformable sleeve for expanding the sleeve radially outwardly. This pressure is greater than the fluid pressure existing in the working cavity between the facing surfaces of the male and female threads and preferably changes as a function of the working pressure in the drilling mud.