  We consider a global, nonlinear version of the Whitney extension problem for
manifold-valued smooth functions on closed domains $C$, with non-smooth
boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with
corners, or is compact and locally convex with rough boundary, we prove that
the restriction map from everywhere-defined functions is a submersion of
locally convex manifolds and so admits local linear splittings on charts. This
is achieved by considering the corresponding restriction map for locally convex
spaces of compactly-supported sections of vector bundles, allowing the even
more general case where $C$ only has mild restrictions on inward and outward
cusps, and proving the existence of an extension operator.
