Let $X$ be a Banach space. By the weak operator topology on $B(X)$, we mean the locally convex topology implemented by the following semi-norms: $$B(X)\to[0,\infty) : T\to|\langle Tx,x^*\rangle|$$ where $x\in X$ and $x^*\in X^*$.

Q. True or false: Assume that with respect to the weak operator topology, the unit ball of $B(X)$ is second countable. Then the Banach space $X^*$ is separable.

I had two abortive attempts:

1st. *Yes it is true:* The wot-separability of $B(X)_{\|.\|\leq1}$ guarantees existence two sequences $\{x_n\}\subseteq X$ and $\{x^*_m\}\subseteq X^*$ such that $N_{n,m}$ forms a sub-basic nbhds (at 0) for the weak operator topology where
$$N_{n,m}=\{T\in B(X)_{\|.\|\leq1}: |\langle Tx_n,x^*_m\rangle|<1\}$$
Let us consider $A$=conv$_r(\{x^*_m\})$, the rational convex hull of $\{x^*_m\}$, which is a countable set. If $A$ is not dense in $X^*$ then one may find $\phi\in X^{**}$ with $\phi(x^*_m)=0$ for all $m\geq1$. It implies that all bounded linear maps of the form of $f\otimes\phi\in B(X,X^{**})$, given by $x\to f(x)\phi$, are contained in the intersection $\bigcap N_{n,m}$. Note that there is no nonzero bounded linear map $T\in B(X)$ contained in the intersection $\bigcap N_{n,m}$but likely there are some operators in $B(X,X^{**})$ contained in the intersection $\bigcap N_{n,m}$!

2ed. To find a counterexample, let us put $X=\ell^1$. We have in general $B(X,Y^{*})\simeq (X\hat{\otimes}Y)^*$. Therefore $B(\ell^1)=(\ell^1\hat{\otimes}c_0)^*$. One may check easily that the inclusion $\iota: B(\ell^1)\to(\ell^1\hat{\otimes}c_0)^*$ is wot-weak star continues but not homeomorphism (to see this, it is enough to consider shifts $T_n(e_k)=e_{n+k}$). Note that the unit ball of $(\ell^1\hat{\otimes}c_0)^*$ is $w^*$-compact and metrizable. Since the inclusion is not hoemorphism, one may not conclude the unit ball of $B(\ell^1)$ is wot-compact and metrizable but probably it remains second countable at least!

Finally note if $X^*$ is separable then the unit ball of $B(X)$ is wot-second countable metrizable space. To see this assertion, note that the projective tensor product $X\hat{\otimes}X^*$ is also separable. Therefore the unit bal of dual space $(X\hat{\otimes}X^*)^*$ is weak-star compact and metrizable. Since the inclusion from ($B(X)_{\|.\|\leq1}$,wot) into the unit ball of $(X\hat{\otimes}X^*)^*$ is a homeomorphism, one may conclude that the unit ball of $B(X)$ is wot-second countable metrizable space.