No. Ady's construction still works, I think. Here's another, easier one:

Choose an orthonormal basis $\{e_{n}\}$ of $H$. Since $\|e_{i} - e_{j}\| = \sqrt{2}$, the continuous functions $f_{n}(x) = \min\{0, 1 - 2 \cdot \|e_{n} - x\|\}$ have disjoint support and sup-norm $1$. This gives an isometric embedding $\ell^{\infty} \to C_{b}(B)$ by sending a bounded sequence $(a_{n})$ to $\sum a_{n} f_{n}$.

A simple modification of this argument shows that $C_{b}(X)$ contains a copy of $\ell^{\infty}$ (and thus isn't separable) whenever the completely regular space $X$ has a countable discrete subset.

No. Ady's construction still works, I think. Here's another, easier one:

Choose an orthonormal basis $\{e_{n}\}$ of $H$. Since $\|e_{i} - e_{j}\| = \sqrt{2}$, the continuous functions $f_{n}(x) = \max\{0, 1 - 2 \cdot \|e_{n} - x\|\}$ have disjoint support and sup-norm $1$. This gives an isometric embedding $\ell^{\infty} \to C_{b}(B)$ by sending a bounded sequence $(a_{n})$ to $\sum a_{n} f_{n}$.

A simple modification of this argument shows that $C_{b}(X)$ contains a copy of $\ell^{\infty}$ (and thus isn't separable) whenever the completely regular space $X$ has a countable discrete subset.