We demonstrate nonadiabatic charge pumping utilizing a sequence of coherent oscillations between a superconducting island and two reservoirs. The pumping rate for each elementary cycle is limited by the coupling between the island and the reservoirs given by the Josephson energy. Our experimental and theoretical studies show that relaxation can be employed to reset the pump in order to avoid accumulation of errors due to nonideal control pulses. Thus our results demonstrate the effects of nonadiabatic quantum pumping and dissipation.