  A subset of a Banach space is called equilateral if the distances between any
two of its distinct elements are the same. It is proved that there exist
non-separable Banach spaces (in fact of density continuum) with no infinite
equilateral subset. These examples are strictly convex renormings of
$\ell_1([0,1])$. A wider class of renormings of $\ell_1([0,1])$ which admit no
uncountable equilateral sets is also considered.
