The kernel of the $\mathbb{ Z}_2^{ n}$ Dirac-Dunkl operator is examined. The symmetry algebra $\mathcal{ A}_{ n}$ of the associated Dirac-Dunkl equation on $\mathbb{ S}^{ n-1}$ is determined and is seen to correspond to a higher rank generalization of the Bannai-Ito algebra. A basis for the polynomial null-solutions of the Dirac-Dunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of $\mathcal{ A}_{ n}$ and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering operators. A scalar realization of $\mathcal{ A}_{ n}$ is proposed.
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