  We study several properties and applications of the ultrapower $M_{\mathcal
U}$ of a metric space $M$. We prove that the Lipschitz-free space $\mathcal
F(M_{\mathcal U})$ is finitely representable in $\mathcal F(M)$. We also
characterize the metric spaces that are finitely Lipschitz representable in a
Banach space as those that biLipschitz embed into an ultrapower of the Banach
space. Thanks to this link, we obtain that if $M$ is finitely Lipschitz
representable in a Banach space $X$, then $\mathcal F(M)$ is finitely
representable in $\mathcal F(X)$. We apply these results to the study of cotype
in Lipschitz-free spaces and the stability of Lipschitz-free spaces and spaces
of Lipschitz functions under ultraproducts.
