Q1. 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 $f\in F$. 

Q2. If it is impossible, what about if we replace $X$ by a separable topological vector space?