  We prove that given an $n$-dimensional integral current space and a
$1$-Lipschitz map, from this space onto the $n$-dimensional Euclidean ball,
that preserves the mass of the current and is injective on the boundary, then
the map has to be an isometry. We deduce as a consequence a stability result
with respect to the intrinsic flat distance, which implies the stability of the
positive mass theorem for graphical manifolds as originally formulated by
Huang--Lee--Sormani.
