AbstractCoexisting density-wave and superconducting states along with the large anomalous Hall effect in the absence of local magnetism remain intriguing and enigmatic features of the AV₃Sb₅ kagome metals (A = K, Rb, Cs). Here, we demonstrate via optical spectroscopy and density-functional calculations that low-energy dynamics of KV₃Sb₅ is characterized by unconventional localized carriers, which are strongly renormalized across the density-wave transition and indicative of electronic correlations. Strong phonon anomalies are prominent not only below the density-wave transition, but also at high temperatures, suggesting an intricate interplay of phonons with the underlying electronic structure. We further propose the star-of-David and tri-hexagon (inverse star-of-David) configurations for the density-wave order in KV₃Sb₅. These configurations are strongly reminiscent of p-wave states expected in the Hubbard model on the kagome lattice at the filling level of the van Hove singularity. The proximity to this regime should have intriguing and far-reaching implications for the physics of KV₃Sb₅ and related materials.