This processing machine comprises a casing (4), a tool-holder spindle (6) which is to carry a machine tool, a positive feed drive mechanism for the spindle (6), which mechanism is suitable for driving the spindle (6) simultaneously in rotation and in translation relative to the casing (4) in accordance with an advance or return movement, and a system (12) for lubricating the machine tool, comprising a lubricant inlet (40), a lubricant duct (42) delimited by the spindle (6), the lubricant duct (42) opening out at an end of the spindle (6) carrying the machine tool, in order to supply the machine tool (7) with lubricant, and a lubricant-admission module (44) connecting the lubricant inlet (40) to the said lubricant duct (42). The lubricant-admission module (44) forms a lubricant pump drawing in the lubricant from the lubricant inlet (40) and supplying the lubricant duct (42) of the spindle (6), the advance and return of the spindle (6) activating the pumping of lubricant by the admission module (44).