The San Andreas Fault Observatory at Depth (SAFOD) scientific borehole near Parkfield, California crosses two actively creeping shear zones at a depth of 2.7 km. Core samples retrieved from these active strands consist of a foliated, Mg-clay-rich gouge containing porphyroclasts of serpentinite and sedimentary rock. The adjacent damage zone and country rocks are comprised of variably deformed, fine-grained sandstones, siltstones, and mudstones. We conducted laboratory tests to measure the permeability of representative samples from each structural unit at effective confining pressures, P_e up to the maximum estimated in situ P_e of 120 MPa. Permeability values of intact samples adjacent to the creeping strands ranged from 10⁻¹⁸ to 10⁻²¹ m² at P_e = 10 MPa and decreased with applied confining pressure to 10⁻²⁰–10⁻²² m² at 120 MPa. Values for intact foliated gouge samples (10⁻²¹–6 × 10⁻²³ m² over the same pressure range) were distinctly lower than those for the surrounding rocks due to their fine-grained, clay-rich character. Permeability of both intact and crushed-and-sieved foliated gouge measured during shearing at P_e ≥ 70 MPa ranged from 2 to 4 × 10⁻²² m² in the direction perpendicular to shearing and was largely insensitive to shear displacement out to a maximum displacement of 10 mm. The weak, actively-deforming foliated gouge zones have ultra-low permeability, making the active strands of the San Andreas Fault effective barriers to cross-fault fluid flow. The low matrix permeability of the San Andreas Fault creeping zones and adjacent rock combined with observations of abundant fractures in the core over a range of scales suggests that fluid flow outside of the actively-deforming gouge zones is probably fracture dominated.