  We prove dynamical stability of a natural class of hypersurface laminations
defined over Cartan--Hadamard manifolds of pinched curvature. We achieve this
by providing a complete solution to the asymptotic Plateau problem for immersed
surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds,
proposed by Labourie in [25], together with its natural higher-dimensional
generalisation.
