  We establish a local-in-space short-time smoothing effect for the
Navier-Stokes equations in the half space. The whole space analogue, due to Jia
and \v{S}ver\'ak [J\v{S}14], is a central tool in two of the authors' recent
work on quantitative $L^3_x$ blow-up criteria [BP21]. The main difficulty is
that the non-local effects of the pressure in the half space are much stronger
than in the whole space. As an application, we demonstrate that the critical
$L^3_x$ norm must concentrate at scales $\sim \sqrt{T^* - t}$ in the presence
of a Type I blow-up.
