  For fixed $k$ we prove exponential lower bounds on the equilateral number of
subspaces of $\ell_{\infty}^n$ of codimension $k$. In particular, we show that
if the unit ball of a normed space of dimension $n$ is a centrally symmetric
polytope with at most $\frac{4n}{3}-o(n)$ pairs of facets, then it has an
equilateral set of cardinality at least $n+1$. These include subspaces of
codimension $2$ of $\ell_{\infty}^{n+2}$ for $n\geq 9$ and of codimension $3$
of $\ell_{\infty}^{n+3}$ for $n\geq 15$.
