Active faults in the upper crust can either slide steadily by aseismic creep, or abruptly causing earthquakes. Creep relaxes the stress and prevents large earthquakes from occurring. Identifying the mechanisms controlling creep, and their evolution with time and depth, represents a major challenge for predicting the behavior of active faults. Based on microstructural studies of rock samples collected from the San Andreas Fault Observatory at Depth (California), we propose that pressure solution creep, a pervasive deformation mechanism, can account for aseismic creep. Experimental data on minerals such as quartz and calcite are used to demonstrate that such creep mechanism can accommodate the documented 20 mm/yr aseismic displacement rate of the San Andreas fault creeping zone. We show how the interaction between fracturing and sealing controls the pressure solution rate, and discuss how such a stress-driven mass transfer process is localized along some segments of the fault.