Does there exist a perfect and star Lindelöf space which is not separable? Does there exist a perfect and star Lindelöf space which is not separable?

A topological space in which every closed set is a $G_\delta$-set is
  called a perfect space.
A topological space $X$ is said to
  be star Lindelöf, if for any open cover $\mathcal U$ of $X$
  there is a Lindelöf subspace $A \subset X$ such that
  $\operatorname{St}(A, \mathcal U)=X$. 

Thanks for your help.

There are certainly counterexamples: all $L$-spaces will do (an $L$-space is a hereditarily Lindelöf regular space, which is not separable),  Such spaces are perfect (because open sets are $F_\sigma$, and so closed sets are $G_\delta$), and trivially star Lindelöf (as any Lindelöf space is).
Moore constructed an $L$-space in ZFC (I now recall).
