Some embodiments are directed to a positive displacement pump having an eccentric piston, comprising a tube having a first end and a second end that is terminated by a cylinder secured to a delivery zone, the tube including an intake opening and a delivery opening, a drive shaft extending between the transmission zone and the tube, a piston arranged in the delivery zone and mounted in a sliding manner at the end of the shaft, being pressed against the cylinder by an elastic presser so as to prevent fluid displacement between the tube and the delivery zone when the pump is dry, and the elastic presser is provided to press the piston against the cylinder when the pump is running under load.