It was shown by A.K. Misra in **"A topological view of $P$-spaces"** that any regular $\aleph_1$-Lindelöf $P$-space is normal (see Corollary 4.6). Also, it is not hard to see that any linearly Lindelöf space must be $\aleph_1$-Lindelöf.

Finding a normal linearly Lindelöf non-Lindelöf space is a very well known open problem in general topology. So now I´m pretty sure that your question is also open.

**Edit:** As pointed out by Robson (see comment below), the notion of $\aleph_1$-Lindelöf used by Misra in his paper doesn't correspond to modern terminology. So I don´t know if a regular linearly Lindelöf $P$-space must be normal. However the following is question 486 in **"Open problems in topology II"**:

(Arhangel´ski) Is the product of two linearly Lindelöf $P$-spaces Lindelöf?

So I still think that the OP´s question is open, since product of two Lindelöf $P$-spaces is Lindelöf (so it is the same question after all).