In this paper, we consider the approximate controllability of semilinear differential inclusion featuring multivalued nonlinearities with non-convex values and the nonlocal conditions described by a multivalued map. Initially, we explore the existence result by assuming the underlying state space to be a reflexive Banach space. Then, we discuss the approximate controllability results in super-reflexive state spaces based on two different growth conditions on the multivalued nonlinearity. Incorporating multivalued maps with non-convex values adds complexity to the studied differential inclusions yet enhances their practical applicability, serving as the core motivation for this research endeavor. We conclude by presenting an illustrative example that satisfies the criteria outlined in this paper.