  Let $X$ be a compact metric space and $\mathcal M_X$ be the set of isometry
classes of compact metric spaces $Y$ such that the Lipschitz distance
$d_L(X,Y)$ is finite. We show that $(\mathcal M_X, d_L)$ is not separable when
$X$ is a closed interval, or an infinite union of shrinking closed intervals.
