A pump having least one pump element with a pump piston guided in a sealed fashion in a cylinder bore (26) of a housing part delimiting a pump working chamber in the cylinder bore, and driven into a stroke motion. The working chamber is connected to an inlet conduit through by an inlet valve opening into the pump working chamber and to an outlet conduit through an outlet valve opening out from the working chamber. The inlet valve has a valve element acted on in a closing direction by a valve spring. The inlet conduit includes a first blind bore which adjoins the pump working chamber and has a smaller diameter than the cylinder bore a valve seat is formed at the transition from the cylinder bore to the first blind bore, and a second blind bore, as another part of the inlet conduit, feeds into the first blind bore.