  We highlight a connection between Diophantine approximation and the lower
Assouad dimension by using information about the latter to show that the
Hausdorff dimension of the set of badly approximable points that lie in certain
non-conformal fractals, known as self-affine sponges, is bounded below by the
dynamical dimension of these fractals. In particular, for self-affine sponges
with equal Hausdorff and dynamical dimensions, the set of badly approximable
points has full Hausdorff dimension in the sponge. Our results, which are the
first to advance beyond the conformal setting, encompass both the case of
Sierpi\'nski sponges/carpets (also known as Bedford-McMullen sponges/carpets)
and the case of Bara\'nski carpets. We use the fact that the lower Assouad
dimension of a hyperplane diffuse set constitutes a lower bound for the
Hausdorff dimension of the set of badly approximable points in that set.
