We provide the explicit formula for the numerical index of any
2-dimensional Lipschitz-free space, also giving the construction of operators
attaining this value as its numerical radius. As a consequence, the numerical
index of 2-dimensional Lipschitz-free spaces can take any value of the
interval [21β,1], and this whole range of numerical indices can be
attained by taking 2-dimensional subspaces of any Lipschitz-free space of the
form F(A), where AβRn with nβ₯2 is any
set with non-empty interior