We search for energetic electron recoil signals induced by boosted dark matter (BDM) from the galactic center using the COSINE-100 array of NaI(Tl) crystal detectors at the Yangyang Underground Laboratory. The signal would be an excess of events with energies above 4 MeV over the well-understood background. Because no excess of events are observed in a 97.7 kg·yr exposure, we set limits on BDM interactions under a variety of hypotheses. Notably, we explored the dark photon parameter space, leading to competitive limits compared to direct dark photon search experiments, particularly for dark photon masses below 4 MeV and considering the invisible decay mode. Furthermore, by comparing our results with a previous BDM search conducted by the Super-Kamionkande experiment, we found that the COSINE-100 detector has advantages in searching for low-mass dark matter. This analysis demonstrates the potential of the COSINE-100 detector to search for MeV electron recoil signals produced by the dark sector particle interactions.