  This paper is concerned with the differential sensitivity analysis of
variational inequalities in Banach spaces whose solution operators satisfy a
generalized Lipschitz condition. We prove a sufficient criterion for the
directional differentiability of the solution map that turns out to be also
necessary for elliptic variational inequalities in Hilbert spaces (even in the
presence of asymmetric bilinear forms, nonlinear operators and nonconvex
functionals). In contrast to classical results, our method of proof does not
rely on Attouch's theorem on the characterization of Mosco convergence but is
fully elementary. Moreover, our technique allows us to also study those cases
where the variational inequality at hand is not uniquely solvable and where
directional differentiability can only be obtained w.r.t. the weak or the
weak-$\star$ topology of the underlying space. As tangible examples, we
consider a variational inequality arising in elastoplasticity, the projection
onto prox-regular sets, and a bang-bang optimal control problem.
