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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 ofthe unit ball containing $x$) contains an element that is almost at distance 2from $x$. We prove the equivalence of Daugavet- and $\Delta$-points in spacesof Lipschitz functions over proper metric spaces and provide twocharacterizations for them. Furthermore, we show that in some spaces ofLipschitz functions, there exist $\Delta$-points that are not Daugavet-points.Lastly, we prove that every space of Lipschitz functions over an infinitemetric space contains a $\Delta$-point but might not contain anyDaugavet-points.

Daugavet- and Delta-points in spaces of Lipschitz functions