We consider nonlinear mappings f : X → Y between Banach spaces and study the notion of restrictive metric regularity of f around some point x ¯ , that is, metric regularity of f from X into the metric space E = f ( X ) . Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves theorem in the case when f is strictly differentiable at x ¯ but its strict derivative ∇ f ( x ¯ ) is not surjective. We develop applications of the results obtained and some other techniques in variational analysis to generalized differential calculus involving normal cones to nonsmooth and nonconvex sets, coderivatives of set-valued mappings, as well as first-order and second-order subdifferentials of extended real-valued functions.