The rheological properties of liquids confined to nanometer scales are important in many physical situations. In this Letter, we demonstrate that the long-range elastic deformation of the confining surfaces must be taken into account when considering the rheology of nanometric liquids. In the case of a squeeze-flow geometry, we show that below a critical distance D(c), the liquid is clamped by its viscosity and its intrinsic properties cannot be disentangled from the global system response. Using nanorheology experiments, we demonstrate that picometer elastic deflections of the rigid confining surfaces dominate the overall mechanical response of nanometric liquids confined between solid walls.