In this paper, the use of hp‐basis functions with higher differentiability properties is discussed in the context of the finite cell method and numerical simulations on complex geometries. For this purpose, Ck hp‐basis functions based on classical B‐splines and a new approach for the construction of C1 hp‐basis functions with minimal local support are introduced. Both approaches allow for hanging nodes, whereas the new C1 approach also includes varying polynomial degrees. The properties of the hp‐basis functions are studied in several numerical experiments, in which a linear elastic problem with some singularities is discretized with adaptive refinements. Furthermore, the application of the Ck hp‐basis functions based on B‐splines is investigated in the context of nonlinear material models, namely hyperelasticity and elastoplasicity with finite strains.