A search for boosted dark matter using 161.9 kt yr of Super-Kamiokande IV data is presented. We search for an excess of elastically scattered electrons above the atmospheric neutrino background, with a visible energy between 100 MeV and 1 TeV, pointing back to the Galactic center or the Sun. No such excess is observed. Limits on boosted dark matter event rates in multiple angular cones around the Galactic center and Sun are calculated. Limits are also calculated for a baseline model of boosted dark matter produced from cold dark matter annihilation or decay. This is the first experimental search for boosted dark matter from the Galactic center or the Sun interacting in a terrestrial detector.