  Let $(M_1,\textit{g}^{(1)})$, $(M_2,\textit{g}^{(2)})$ be closed Riemannian
spin manifolds. We study the existence of solutions of the spinorial Yamabe
problem on the product $M_1\times M_2$ equipped with a family of metrics
$\varepsilon^{-2}\textit{g}^{(1)}\oplus\textit{g}^{(2)}$, $\varepsilon>0$. Via
variational methods and blow-up techniques, we prove the existence of solutions
which depend only on the factor $M_1$, and which exhibit a spike layer as
$\varepsilon\to0$. Moreover, we locate the asymptotic position of the peak
points of the solutions in terms of the curvature tensor on
$(M_1,\textit{g}^{(1)})$.
