A dispersive-optical-model analysis of p+ 40,42,44,48 Ca and n+ 40 Ca interactions has been carried out. The real and imaginary potentials have been constrained by fitting elastic-scattering data, total and reaction cross sections, and level properties of valence hole states deduced from (e, e p) data. The resulting surface imaginary potential increases with asymmetry for protons, implying that in heavier Ca isotopes, protons experience stronger long-range correlations. Presently, there is not enough data for neutrons to determine their asymmetry dependence. Global optical-model fits usually assume that the neutron asymmetry dependence is equal in magnitude, but opposite in sign, to that for protons. Such a dependence was found to give unphysical results for heavy Ca isotopes. The dispersive optical model is shown to be a useful tool for data-driven extrapolations to the drip lines. Neutron and proton data at larger asymmetries are needed to achieve more reliable extrapolations. The present analysis predicts 60 Ca and 70 Ca to be bound, while the intermediate isotopes are not.