The answer to the title question is given in terms of the elementary properties of Banach spaces regarded as structures for a certain first-order language. The same question for Banach space ultrapowers is also considered. The connection between nonstandard hulls and Banach space ultrapowers derives in part from the following fact, of independent interest in nonstandard analysis: for each cardinal number κ there exist ultrapower enlargements which are κ-saturated and which have the κ-isomorphism property.