We report on a highly-efficient experimental scheme for the generation of deep-ultraviolet ultrashort light pulses using four-wave mixing in gas-filled kagomé-style photonic crystal fiber. By pumping with ultrashort, few μJ, pulses centered at 400 nm, we generate an idler pulse at 266 nm, and amplify a seeded signal at 800 nm. We achieve remarkably high pump-to-idler energy conversion efficiencies of up to 38%. Although the pump and seed pulse durations are ~100 fs, the generated ultraviolet spectral bandwidths support sub-15 fs pulses. These can be further extended to support few-cycle pulses. Four-wave mixing in gas-filled hollow-core fibres can be scaled to high average powers and different spectral regions such as the vacuum ultraviolet (100-200 nm).