A norm one element x of a Banach space is a Daugavet-point (respectively, a \(\Delta \) -point) if every slice of the unit ball (respectively, every slice of the unit ball containing x) contains an element that is almost at distance 2 from x. We prove the equivalence of Daugavet- and \(\Delta \) -points in spaces of Lipschitz functions over proper metric spaces and provide two characterizations for them. We also show that every space of Lipschitz functions over an unbounded or not uniformly discrete metric space contains a Daugavet-point and every space of Lipschitz functions over an infinite metric space contains a \(\Delta \) -point. Lastly, we show that there exists an infinite metric space such that the corresponding space of Lipschitz functions does not contain any Daugavet-points, thus also proving that Daugavet- and \(\Delta \) -points do not always coincide in spaces of Lipschitz functions.
