For a classical contact problem with wear, we consider a newly proposed model where a relation between the pressure and the wear rate is due to a combination of fractional calculus and relaxation effects. The stationary pressure profile and the rate of stabilisation are studied for practically interesting cases of constant and oscillatory loads in time. Numerical simulations reveal differences in a qualitative behavior depending on the model parameters. The proposed model is expected to serve as a simple solvable model for capturing complicated phenomena appearing in complex contact scenarios.