  We present a quasi-local version of the stability of the positive mass
theorem. We work with the Brown--York quasi-local mass as it possesses
positivity and rigidity properties, and therefore the stability of this
rigidity statement can be studied. Specifically, we ask if the Brown--York mass
of the boundary of some compact manifold is close to zero, must the manifold be
close to a Euclidean domain in some sense?
  Here we consider a class of compact $n$-manifolds with boundary that can be
realized as graphs in $\mathbb{R}^{n+1}$, and establish the following. If the
Brown--York mass of the boundary of such a compact manifold is small, then the
manifold is close to a Euclidean hyperplane with respect to the
Federer--Fleming flat distance.
