We consider a non-standard boundary value problem characterizing deformations of a composite consisting of a arbitrarily-shaped elastic inclusion embedded in an infinite elastic matrix subjected to uniform remote stresses. The interface between the inclusion and the surrounding matrix is taken to be imperfect with ’spring-like’ interface parameters describing the properties of the interface layer. We show that any classical solution to the boundary value problem is necessarily unique despite the fact that the asymptotic behaviour of the solution is not accommodated by the corresponding classical results from the same theory of elasticity.