  A quantum superintegrable model with reflections on the $(n-1)$-sphere is
presented. Its symmetry algebra is identified with the higher rank
generalization of the Bannai-Ito algebra. It is shown that the Hamiltonian of
the system can be constructed from the tensor product of $n$ representations of
the superalgebra $\mathfrak{osp}(1|2)$ and that the superintegrability is
naturally understood in that setting. The separated solutions are obtained
through the Fischer decomposition and a Cauchy-Kovalevskaia extension theorem.
