  We study the moduli space of Yang--Mills connections on bundles over a
conformally compact manifold $\overline{M}$. We prove that, for every
Yang--Mills connection $A$ that satisfies an appropriate nondegeneracy
condition, and for every small deformation $\gamma$ of
$A_{|\partial\overline{M}}$, there is a Yang--Mills connection in the interior
that extends $A_{|\partial\overline{M}}+\gamma$. As a corollary, we confirm an
expectation of Witten mentioned in his foundational paper about holography
[arXiv:hep-th/9802150].
