Daugavet- and Delta-points in spaces of Lipschitz functions

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. Furthermore, we show that in some spaces of Lipschitz functions, there exist $\Delta$-points that are not Daugavet-points. Lastly, we prove that every space of Lipschitz functions over an infinite metric space contains a $\Delta$-point but might not contain any Daugavet-points.