$\newcommand{\P}{\mathcal{P}}$

Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) metric on $\P(E)$.

It is well known that if additionally the space $E$ is *separable*, then the space $(\P(E),W_d)$ is complete. It is also known that in the general case (if $E$ is not separable) only the space $(\P_r(E),W_d)$ is complete, where $\P_r(E)$ is the set of all Radon probability measures.

My question is: can one construct an explicit example of a complete (and obviously non-separable) metric space$(E,d)$ such that the space $(\P(E),W_d)$ is not complete?