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Gabber deduced his theorem of independence of $l$ of intersection cohomologyfrom a general stability result over finite fields. In this article, we provean analogue of this general result over local fields. More precisely, weintroduce a notion of independence of $l$ for systems of complexes of $l$-adicsheaves on schemes of finite type over a local field, equivariant under finitegroups. We establish its stability by Grothendieck's six operations and thenearby cycle functor. Our method leads to a new proof of Gabber's theorem. Wealso give a generalization to algebraic stacks.    Gabber a d\'eduit son th\'eor\`eme d'ind\'ependance de $l$ de la cohomologiel'intersection d'un r\'esultat g\'en\'eral de stabilit\'e sur les corps finis.Dans cet article, nous d\'emontrons un analogue sur les corps locaux de cer\'esultat g\'en\'eral. Plus pr\'ecis\'ement, nous introduisons une notiond'ind\'ependance de $l$ pour les syst\`emes de complexes de faisceaux$l$-adiques sur les sch\'emas de type fini sur un corps local \'equivariantssous des groupes finis et nous \'etablissons sa stabilit\'e par les sixop\'erations de Grothendieck et le foncteur des cycles proches. Notre m\'ethodepermet d'obtenir une nouvelle d\'emonstration du th\'eor\`eme de Gabber. Nousdonnons aussi une g\'en\'eralisation aux champs alg\'ebriques.

Sur l'ind\'ependance de l en cohomologie l-adique sur les corps locaux