The answer depends on what you mean by a closed manifold and a compact operator. If the manifold is compact without boundary then the spaces involved are Banach spaces and the answer is yes and this is indeed proved using the Arzela-Ascoli theorem, after a standard localisation argument.
If the manifold is without boundary but not compact (think real line), then the spaces are (nuclear) Frechet spaces and the answer depends on what you mean by a compact operator on a locally convex space. If you define these as taking bounded sets into relatively compact ones, then the answer is yes, but if you mean those which take a neighbourhood of zero to a relativeley compact set (the usual definition), then the answer is no.
