Does separability imply the Lindelöf property?

Does separability imply a sort of Lindelöf property?

Since I can't prove this fact I'm beginning to think that my conjecture is false.

Intuitively, $\mathbb{R}$ has a countable subset $\mathbb{Q}$ which is used to form a countable basis for $\mathbb{R}$ with the usual topology and prove the Lindeöf property.

Counterexamples in Topology lists several spaces which are separable but not Lindelöf. You can generate a list at Spacebook.

A nice example that is useful to know is the Mrówka space $\Psi$, which among other things is Tikhonov, separable, pseudocompact, not countably compact, and not Lindelöf. I also described it briefly in this answer, where I gave an example of a specific open cover with no countable subcover. This answer contains a more complete description of the construction of the space, and this one has another way to get the almost disjoint sets that are needed.