We have studied the onset of plasticity in coherently-strained semiconductor superlattices, using nano-indentation with spherical indenter tips to observe the full stress-strain curve. The yield pressure is reduced by as much as a factor of two by the presence of the coherency strain. By varying the thicknesses and strains of the superlattice layers, we provide a proof that yield commences over a finite volume. It is properties averaged or summed over this volume which determine the yield pressure. We show that the relevant yield criterion for our experimental data is the rate of change of elastic strain energy with plastic relaxation, integrated over a volume of the order of a micron across. This result is expected to be valid for other systems with highly inhomogenous strain fields, and hence to be applicable to modelling of point contact, and to the design and understanding of structural materials which have coherently-strained microstructure.