  A complex manifold Y satisfies the Convex Approximation Property (CAP) if
every holomorphic map from a neighborhood of a compact convex set K in a
complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire
maps from C^n to Y. If X is a reduced Stein space and Z is a stratified
holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections
of Z over X enjoy the Oka property with (jet) interpolation and approximation.
Previously this has been proved by the author in the case when X is a Stein
manifold without singularities (Ann. Math., 163 (2006), 689-707,
math.CV/0402278; Ann. Inst. Fourier, 55 (2005), 733-751, math.CV/0411048). We
also give existence results for holomorphic sections under certain connectivity
hypothesis on the fibers. In the final part of the paper we obtain the Oka
property for sections of submersions with stratified sprays over Stein spaces.
