  For every $f \in L^N(\Omega)$ defined in an open bounded subset $\Omega$ of
$\mathbb{R}^N$, we prove that a solution $u \in W_0^{1, 1}(\Omega)$ of the
$1$-Laplacian equation ${-}\mathrm{div}{(\frac{\nabla u}{|\nabla u|})} = f$ in
$\Omega$ satisfies $\nabla u = 0$ on a set of positive Lebesgue measure. The
same property holds if $f \not\in L^N(\Omega)$ has small norm in the
Marcinkiewicz space of weak-$L^{N}$ functions or if $u$ is a BV minimizer of
the associated energy functional. The proofs rely on Stampacchia's truncation
method.
