  We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for
asymptotically cylindrical K\"ahler manifolds: If $\mathscr{E}$ is a reflexive
sheaf over an ACyl K\"ahler manifold, which is asymptotic to a $\mu$-stable
holomorphic vector bundle, then it admits an asymptotically
translation-invariant protectively Hermitian Yang-Mills metrics (with curvature
in $L^2_{\mathrm{loc}}$ across the singular set). Our proof combines the
analytic continuity method of Uhlenbeck and Yau [UY86] with the geometric
regularization scheme introduced by Bando and Siu [BS94].
