Conditions for a process \(\xi\) on a compact metric space \({\mathcal S}\) to be simultaneously max-infinitely divisible and sample continuous are obtained. Although they fall short of a complete characterization of such processes, these conditions yield complete descriptions of the sample continuous non-degerate max-stable processes on \({\mathcal S}\) and of the infinitely divisible non-void random compact subsets of a Banach space under the operation of convex hull of unions.