Let $(X,d)$ be a separable and connected metric space.  My question is rather short and to the point: do there exist $\{x_n\}_{n=0}^{\infty}\subseteq X$ such that
$$
\left\{d(x_n,\cdot)-d(x_0,\cdot)\right\}_{n=1}^{\infty},
$$
is a Schauder basis of $C_b(X)$?  If so, what is this "basis" called in the literature?