A completely regular topological space $(X,\tau)$ is called a $P$-space, if every $G_\delta$-subset of $X$ is open .$($i.e $\tau$ is closed under countable intersection$)$. Here we recall some special properties of $P$-spaces:
Every countable subset of $X$ is obviously closed and discrete.
Every countable subset of $X$ is $C$-embedded in $X$.$($i.e. every continuous real valued function on a countable subset of $X$ can be extended to all of $X$ $)$
Now with the sake of above properties I could pose my Questions. My questions that are given as follows are the extended form of these properties of countable sets to Lindelöf subsets of $P$-spaces.
Is it true that in every $P$-space, every Lindelöf subset is closed?
Is it true that in every $P$-space every Lindelöf subset is $C$-embedded in $X$?