  Persistence of non-degeneracy is a phenomenon which appears in the theory of
$\overline{\mathbb Q}_l$-representations of the linear group: every irreducible
submodule of the restriction to the mirabolic sub-representation of a
non-degenerate irreducible representation is non-degenerate. This is not true
anymore in general, if we look at the modulo $l$ reduction of some stable
lattice. As in the Clozel-Harris-Taylor generalization of global Ihara's lemma,
we show that this property, called non-degeneracy persistence and related to
the notion of essentially absolutely irreducible and generic representations in
the work of Emerton-Helm, remains true for lattices given by the cohomology of
Lubin-Tate spaces. As an global application, we give a new construction of
automorphic congruences in the Ribet spirit.

Number Theory