  The border-collision normal form is a piecewise-linear continuous map on
$\mathbb{R}^N$ that describes dynamics near border-collision bifurcations of
nonsmooth maps. This paper studies a codimension-three scenario at which the
border-collision normal form with $N=2$ exhibits infinitely many attracting
periodic solutions. In this scenario there is a saddle-type periodic solution
with branches of stable and unstable manifolds that are coincident, and an
infinite sequence of attracting periodic solutions that converges to an orbit
homoclinic to the saddle-type solution. Several important features of the
scenario are shown to be universal, and three examples are given. For one of
these examples infinite coexistence is proved directly by explicitly computing
periodic solutions in the infinite sequence.
