The goal of this paper is to study geometric and extremal properties of the convex body B F (M) , which is the unit ball of the Lipschitz-free Banach space associated with a finite metric space M. We investigate 1 and ∞-sums, in particular we characterize the metric spaces such that B F (M) is a Hanner polytope. We also characterize the finite metric spaces whose Lipschitz-free spaces are isometric. We discuss the extreme properties of the volume product P(M) = |B F (M) | · |B • F (M) |, when the number of elements of M is fixed. We show that if P(M) is maximal among all the metric spaces with the same number of points, then all triangle inequalities in M are strict and B F (M) is simplicial. We also focus on the metric spaces minimizing P(M), and on Mahler's conjecture for this class of convex bodies.