We give new versions of the first and the second deformation theorems in the context of invariant locally Lipschitz maps on a real Banach space endowed with an isometric representation of a compact topological group. As an application of the second deformation result, we define equivariant critical groups for locally Lipschitz maps, extend standard results of Morse theory, and obtain, as a byproduct, existence and multiplicity results for critical orbits, involving in particular a notion of equivariant homological linking of pairs. As an application of the first deformation result, we establish a general minimax principle based on a notion of equivariant homotopical linking of pairs.