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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.

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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).

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    Hadn’t seen that result; thanks for the reference! – 2017-01-14
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    @BrianM.Scott I worked with compact $L$-spaces in the past, which are killed by MA($\omega_1)$, hence my interest in $L$-spaces and for a long time a ZFC $L$-space was open, so I was suprised when Moore came with a "real" example, when $S$-spaces need not exist in ZFC. – 2017-01-15