  The $q$-deformed Bannai-Ito algebra was recently constructed in the threefold
tensor product of the quantum superalgebra $\mathfrak{osp}_q(1\vert 2)$. It
turned out to be isomorphic to the Askey-Wilson algebra. In the present paper
these results will be extended to higher rank. The rank $n-2$ $q$-Bannai-Ito
algebra $\mathcal{A}_n^q$, which by the established isomorphism also yields a
higher rank version of the Askey-Wilson algebra, is constructed in the $n$-fold
tensor product of $\mathfrak{osp}_q(1\vert 2)$. An explicit realization in
terms of $q$-shift operators and reflections is proposed, which will be called
the $\mathbb{Z}_2^n$ $q$-Dirac-Dunkl model. The algebra $\mathcal{A}_n^q$ is
shown to arise as the symmetry algebra of the constructed $\mathbb{Z}_2^n$
$q$-Dirac-Dunkl operator and to act irreducibly on modules of its polynomial
null-solutions. An explicit basis for these modules is obtained using a
$q$-deformed $\mathbf{CK}$-extension and Fischer decomposition.
