  The Dirac-Dunkl operator on the 2-sphere associated to the $\mathbb{Z}_2^3$
reflection group is considered. Its symmetries are found and are shown to
generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are
constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl
operator are obtained using a Cauchy-Kovalevskaia extension theorem. These
eigenfunctions, which correspond to Dunkl monogenics, are seen to support
finite-dimensional irreducible representations of the Bannai-Ito algebra.
