A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersection).

A topological space $X$ is linearly Lindelöf if for every open cover of $X$, linearly ordered by the subset relation, has a countable subcover.

Is there a $P$-space linearly Lindelöf and non-Lindelöf?

A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersections).

A topological space $X$ is linearly Lindelöf if every open cover of $X$ which is linearly ordered by the subset relation has a countable subcover.

Is there a linearly Lindelöf non-Lindelöf $P$-space?