In thin films of mixed ionic electronic conductors sandwiched by two ion-blocking electrodes, the homogeneous migration of ions and their polarization will modify the electronic carrier distribution across the conductor, thereby enabling homogeneous resistive switching. Here we report non-filamentary memristive switching based on the bulk oxide ion conductivity of amorphous GaO x (xB1.1) thin films. We directly observe reversible enrichment and depletion of oxygen ions at the blocking electrodes responding to the bias polarity by using photoemission and transmission electron microscopies, thus proving that oxygen ion mobility at room temperature causes memristive behaviour. The shape of the hysteresis I-V curves is tunable by the bias history, ranging from narrow counter figure-eight loops to wide hysteresis, triangle loops as found in the mathematically derived memristor model. This dynamical behaviour can be attributed to the coupled ion drift and diffusion motion and the oxygen concentration profile acting as a state function of the memristor.