A class of quasi variational–hemivariational inequalities in reflexive Banach spaces is studied. The inequalities contain a convex potential, a locally Lipschitz superpotential and an implicit obstacle set of constraints. Results on the well-posedness including existence, uniqueness, dependence of solution on the data and the compactness of the solution set in the strong topology are established. The applicability of the results is illustrated by the steady-state generalized Stokes model of a generalized Newtonian incompressible fluid with a non-monotone slip boundary condition. This article is part of the theme issue ‘Non-smooth variational problems and applications’.