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We show that the class of Lipschitz-free spaces over closed subsets of anycomplete metric space $M$ is closed under arbitrary intersections, improvingupon the previously known finite-diameter case. This allows us to formulate ageneral and natural definition of supports for elements in a Lipschitz-freespace $\mathcal F(M)$. We then use this concept to study the extremal structureof $\mathcal F(M)$. We prove in particular that $(\delta(x) -\delta(y))/d(x,y)$ is an exposed point of the unit ball of $\mathcal F(M)$whenever the metric segment $[x,y]$ is trivial, and that any extreme pointwhich can be expressed as a finitely supported perturbation of a positiveelement must be finitely supported itself. We also characterise the extremepoints of the positive unit ball: they are precisely the normalized evaluationfunctionals on points of $M$.

Supports in Lipschitz-free spaces and applications to extremal structure