On the German Wikipedia one can read the following about the Arzelà-Ascoli theorem:
The theorem of Arzelà-Ascoli can be generalised to families of equicontinuous maps with values in a compact manifold.
Unfortunately, no reference is given. Does anybody has a reference for this statement?
Every compact Riemannian manifold carries a Riemannian metric. Given such a metric, you can embed it isometrically into $\mathbb{R}^N$ for sufficiently large $N$ by Nash's embedding theorem and apply Arzelà-Ascoli to each of the component functions. Of course, this is a huge overkill.
General versions of the Arzelà-Ascoli theorem that cover the case you mention can be found in most introductory books on general topology, see e.g. Theorem 8.2.10 in Engelking, General Topology.