We demonstrate the generation of wavelength-tunable deep-ultraviolet pulses in a small-mode-area hollow-core fiber fabricated by tapering a nodeless tubular-type hollow-core fiber. Down-scaling of the cross-sectional geometry reduces the pump energy requirement for inducing sufficient nonlinear effects, presenting a unique opportunity for staging low-energy-threshold gas-based nonlinear optics. We report the onset of the ultraviolet light with the pump pulse energy as low as 125 nJ. Our numerical analysis shows that the frequency conversion arises due to soliton phase matching, and therefore shot-to-shot coherence of the ultraviolet emission is well-preserved. It offers a promising platform for a compact ultraviolet frequency comb source.