  We find Weitzenb\"ock formula for the Fueter-Dirac operator which controls
the infinitesimal deformations of an associative submanifold in a $7$--manifold
with a $G_2$--structure. We establish a vanishing theorem to conclude rigidity
under some positivity assumptions on curvature, which are particularly mild in
the nearly parallel case. As applications, we give a different proof of
rigidity for one of Lotay's associatives in the round $7$-sphere from those
given by Kawai. We also provide simpler proofs of previous results by Gayet for
the Bryant-Salamon metric. Finally, we obtain an original example of a rigid
associative in a compact manifold with locally conformal calibrated
$G_2$-structure obtained by Fernandez-Fino-Raffero.
