An approximate solution is obtained for the contact problem of a layer of finite thickness loaded by a rough cylindrical stamp which moves along the boundary. The coefficient of friction is assumed to be constant. The lower side of the layer is attached to a rigid base. In the problem inertial forces are neglected and the solution is approximated by a plane strain solution. This solution is presented in the form of a (convergent) series expansion in powers of the reciprocal thickness parameter, i.e. the ratio of the values of the thickness and the length of the contact region. The coefficients in this expansion satisfy singular integral equations. Numerical results are obtained for large values of the thickness parameter. In part II of this investigation the thin layer asymptotics will be given.