It is often conjectured that the existence and uniqueness of solutions to the quasi-static Signorini problem with Coulomb friction should hold, provided that the friction coefficient is lower than a critical value. Recently, the existence of solutions to the quasi-static Signorini problem with non-local Coulomb friction was shown (M. Cocu, E. Pratt, M. Raous, Int. J. Engng. Sci. 34 (1996) 783–798) in functional spaces of type $W^{1,p}(0,T)$ and for a sufficiently low friction coefficient. In this paper, it is proved that uniqueness does not hold, in general, for an arbitrarily small friction coefficient.