We study theoretically the effect of finite elasticity on the attractive power-law interaction of two solids separated by a thin liquid (or gas) film. A new asymptotic technique is developed to determine the deformed shape of the surfaces and to calculate the elasticity contribution to the total force, i.e., an additional term present between the deformed bodies. Both the deformation and the elasticity contribution are found to be nonnegligible well before contact is reached, although they are of much shorter range than the surface force that caused them. This range can be characterized by a reference elasticity length, which depends on elastic constants and size of the solids, as well as on the attractive force that led to deformation. The total force vs separation profile for elastic surfaces is found to depend on how the measurements are made, namely, how the separation is detected: it can lead to either less or more attractive force compared with the case of rigid surfaces.