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We show that three-dimensional general relativity, augmented with two vector fields, allows for a nonrelativistic limit, different from the standard limit leading to Newtonian gravity, that results in a well-defined action which is of the Chern-Simons type. We show that this three-dimensional "extended Bargmann gravity," after coupling to matter, leads to equations of motion allowing a wider class of background geometries than the ones that one encounters in Newtonian gravity. We give the supersymmetric generalization of these results and point out an important application in the context of calculating partition functions of nonrelativistic field theories using localization techniques.