  We study a class of asymptotically cylindrical Ricci-flat K\"ahler metrics
arising on quasiprojective manifolds. Using the Calabi--Yau geometry and
analysis and the Kodaira--Kuranishi--Spencer theory and building up on results
of N.Koiso for the case of compact manifolds, we show that under rather general
hypotheses any `small' asymptotically cylindrical Ricci-flat deformations of
asymptotically cylindrical Ricci-flat K\"ahler metrics are again K\"ahler,
possibly with respect to a perturbed complex structure. We also find the
dimension of the moduli space for these small deformations. In the class of
asymptotically cylindrical Ricci-flat metrics on $2n$-manifolds, the holonomy
reduction to SU(n) is an open condition.
