Characterization of ${\mathrm{\ell }}_p$-like and $c_0$-like equivalence relations

Let $X$ be a Polish space, $d$ a pseudo-metric on $X$. If $\{(u,v):d(u,v)<\delta \}$ is ${\mathbf{\Pi }}_1^1$ for each $\delta >0$, we show that either $(X,d)$ is separable or there are $\delta >0$ and a perfect set $C\subseteq X$ such that $d(u,v)\ge \delta$ for distinct $u,v\in C$. Granting this dichotomy, we characterize the positions of ${\mathrm{\ell }}_p$-like and $c_0$-like equivalence relations in the Borel reducibility hierarchy.