Nonlinear piezoelectricity has recently been detected in strained-layer superlattices where CdTe is pseudomorphically grown along (111) on different materials. We investigate this problem from first principles using density-functional theory. As a byproduct, we provide the ab initio calculation of several properties of this material, including the equilibrium lattice constant, Born effective charges, internal strain parameter, zone-center optical-phonon frequencies, dielectric constant, elastic constants, and the «clamped-ions» piezoelectric constant. Nonlinear effects on all of them are considered. Our major result is that piezoelectricity is accurately linear over a wide range of volume-conserving strains, while it displays strong nonlinearity whenever the strain is not volume conserving