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Sufficiently large electric current applied to metallic nanostructures can bring them far out of equilibrium, resulting in non-Ohmic behaviors characterized by current-dependent resistance. We experimentally demonstrate a linear dependence of resistance on current in microscopic thin-film metallic wires at cryogenic temperatures, and show that our results are inconsistent with common non-Ohmic mechanisms such as Joule heating. As the temperature is increased, the linear dependence becomes smoothed out, resulting in the crossover to behaviors consistent with Joule heating. A plausible explanation for the observed behaviors is the strongly nonequilibrium distribution of phonons generated by the current. Analysis based on this interpretation suggests that the observed anomalous current-dependent resistance can provide information about phonon transport and electron-phonon interaction at nanoscale. The ability to control the properties of phonons generated by current can lead to new routes for the optimization of thermal properties of electronic nanodevices.