We propose a quantum memory protocol based on trapping photons in a fiber-integrated cavity, comprised of a birefringent fiber with dichroic reflective end facets. Photons are switched into resonance with the fiber cavity by intracavity Bragg-scattering frequency translation, driven by ancillary control pulses. After the storage delay, photons are switched out of resonance with the cavity, again by intracavity frequency translation. We demonstrate storage of quantum-level THz-bandwidth coherent states for a lifetime up to 16 cavity round trips, or 200 ns, and a maximum overall efficiency of 73%.