Does there exist a separable Banach space $X$ satisfying in the following property?

1- $X^*$ is non separable.

2- For every countable subset $F\subset X^*$ there exists $0\neq x_F\in X$ such that $f(x_F)=0$ for all $f\in F$.

If it is impossible, what about if we replace $X$ by a separable topological vector space?