Invariant fiber bundles are the generalization of invariant manifolds from discrete dynamical systems (mappings) to non-autonomous difference equations. In this paper we present a self-contained proof of their existence and smoothness. Our main result generalizes the so-called "Hadamard-Perron-Theorem" for time-dependent families of pseudo-hyperbolic mappings from the finite-dimensional invertible to the infinite-dimensional non-invertible case.