In this work we show that the rate-dependent hysteresis seen in current-voltage scans of CH3NH3PbI3 perovskite solar cells is related to a slow field-induced process that tends to cancel the electric field in the device at each applied bias voltage. It is attributed to the build-up of space charge close to the contacts, independent of illumination and most likely due to ionic displacement, which is enhanced when the device undergoes aging. This process can also lead to a reduction of the open-circuit voltage or the steady-state photocurrent and does not directly correlate with the development of the hysteresis if it is measured at a fixed voltage sweep rate.