  We construct finite-range interactions on $\mathcal{S}^{\mathbb{Z}^2}$, where
$\mathcal{S}$ is a finite set, for which the associated equilibrium states
(i.e., the shift-invariant Gibbs states) fail to converge as temperature goes
to zero. More precisely, if we pick any one-parameter family
$(\mu_\beta)_{\beta>0}$ in which $\mu_\beta$ is an equilibrium state at inverse
temperature $ \beta$ for this interaction, then
$\lim_{\beta\to\infty}\mu_\beta$ does not exist. This settles a question posed
by the first author and Hochman who obtained such a non-convergence behavior
when $d\geq 3$, $d$ being the dimension of the lattice.
