  We study Lipschitz-free spaces over compact and uniformly discrete metric
spaces enjoying certain high regularity properties - having group structure
with left-invariant metric. Using methods of harmonic analysis we show that,
given a compact metrizable group $G$ equipped with an arbitrary compatible
left-invariant metric $d$, the Lipschitz-free space over $G$,
$\mathcal{F}(G,d)$, satisfies the metric approximation property. We show also
that, given a finitely generated group $G$, with its word metric $d$, from a
class of groups admitting a certain special type of combing, which includes all
hyperbolic groups and Artin groups of large type, $\mathcal{F}(G,d)$ has a
Schauder basis. Examples and applications are discussed. In particular, for any
net $N$ in a real hyperbolic $n$-space $\mathbb{H}^n$, $\mathcal{F}(N)$ has a
Schauder basis.
