Jets are ubiquitously present in both quiet and active regions on the Sun. They are widely believed to be driven by magnetic reconnection. A fan-spine structure has been frequently reported in some coronal jets and flares, and has been regarded as a signature of ongoing magnetic reconnection in a topology consisting of a magnetic null connected by a fan-like separatrix surface and a spine. However, for small-scale chromospheric jets, clear evidence of such structures is rather rare, although it has been implied in earlier works that showed an inverted-Y-shaped feature. Here we report high-resolution (0.″16) observations of a small-scale chromospheric jet obtained by the New Solar Telescope (NST) using 10830 ̊A filtergrams. Bi-directional flows were observed across the separatrix regions in the 10830 Å images, suggesting that the jet was produced due to magnetic reconnection. At the base of the jet, a fan-spine structure was clearly resolved by the NST, including the spine and the fan-like surface, as well as the loops before and after the reconnection. A major part of this fan-spine structure, with the exception of its bright footpoints and part of the base arc, was invisible in the extreme ultraviolet and soft X-ray images (observed by the Atmosphere Imaging Assembly and the X-Ray Telescope, respectively), indicating that the reconnection occurred in the upper chromosphere. Our observations suggest that the evolution of this chromospheric jet is consistent with a two-step reconnection scenario proposed by Török et al.