We report limits on weakly interacting massive particle (WIMP) dark matter derived from three years of data collected by the COSINE-100 experiment with NaI(Tl) crystals, achieving an improved energy threshold of 0.7 keV. This lowered threshold enhances sensitivity in the sub-GeV mass range, extending the reach for direct detection of low-mass dark matter. Although no excess of WIMP-like events was observed, the increased sensitivity enabled a model-insensitive comparison between the expected WIMP signal rate-based on mass limits from our data-and DAMA's reported modulation amplitude. Our findings strongly disfavor the DAMA signal as originating from WIMP interactions, fully excluding DAMA/LIBRA 3σ allowed regions and providing enhanced WIMP mass limits by an order of magnitude in the spin-independent model compared to previous results. In the spin-dependent model, cross-section upper limits were obtained in the mass range [0.1-5.0] GeV/c^{2}, with additional sensitivity to sub-GeV WIMPs through the inclusion of the Migdal effect. These results represent substantial progress in low-mass dark matter exploration and reinforce constraints on the longstanding DAMA claim.