  Let D be a relatively compact strongly pseudoconvex domain in a Stein
manifold, and let Y be a complex manifold. We prove that the set A(D,Y),
consisting of all continuous maps from the closure of D to Y which are
holomorphic in D, is a complex Banach manifold. When D is the unit disc in C
(or any other topologically trivial strongly pseudoconvex domain in a Stein
manifold), A(D,Y) is locally modeled on the Banach space A(D,C^n)=A(D)^n with
n=dim Y. Analogous results hold for maps which are holomorphic in D and of
class C^r up to the boundary for any positive integer r. We also establish the
Oka property for sections of continuous or smooth fiber bundles over the
closure of D which are holomorphic over D and whose fiber enjoys the Convex
approximation property. The main analytic technique used in the paper is a
method of gluing holomorphic sprays over Cartan pairs in Stein manifolds, with
control up to the boundary, which was developed in our paper "Holomorphic
curves in complex manifolds" (Duke Math. J. 139 (2007), no. 2, 203--253).
