  Let $T$ be a `properly metrisable' topological space, that is, locally compact, separable and metrisable. Let $\mathcal{M}^T$ be the non-empty set of all proper metrics $d$ on $T$ compatible with its topology, and equip $\mathcal{M}^T$ with the topology of uniform convergence, where the metrics are regarded as functions on $T^2$. We prove that if $T$ is uncountable then the set $\mathcal{A}^T_f$ of metrics $d\in\mathcal{M}^T$ for which the Lipschitz-free space $\mathcal{F}(T,d)$ fails the approximation property is a dense set in $\mathcal{M}^T$. Combining this with a result of Dalet, we conclude that for any properly metrisable space $T$, $\mathcal{A}^T_f$ is either empty or dense in $\mathcal{M}^T$. 
Lipschitz-free spaces over properly metrisable spaces and approximation   properties