  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.
