  Boundaries in three-dimensional $\mathcal{N}=2$ superconformal theories may
preserve one half of the original bulk supersymmetry. There are two
possibilities which are characterized by the chirality of the leftover
supercharges. Depending on the choice, the remaining $2d$ boundary algebra
exhibits $\mathcal{N}=(0,2)$ or $\mathcal{N}=(1,1)$ supersymmetry. In this work
we focus on correlation functions of chiral fields for both types of
supersymmetric boundaries. We study a host of correlators using superspace
techniques and calculate superconformal blocks for two- and three-point
functions. For $\mathcal{N}=(1,1)$ supersymmetry, some of our results can be
analytically continued in the spacetime dimension while keeping the codimension
fixed. This opens the door for a bootstrap analysis of the $\epsilon$-expansion
in supersymmetric BCFTs. Armed with our analytically-continued superblocks, we
prove that in the free theory limit two-point functions of chiral (and
antichiral) fields are unique. The first order correction, which already
describes interactions, is universal up to two free parameters. As a check of
our analysis, we study the Wess-Zumino model with a supersymmetric boundary
using Feynman diagrams, and find perfect agreement between the perturbative and
bootstrap results.
