Is each paracompact space with countable cellularity always Lindelöf? Is each paracompact space with countable cellularity always Lindelöf?

We recall that the cellularity of a space is the minimal infinite cardinal $\kappa$ such that every family of pairwise disjoint open sets has cardinality less than or equal to $\kappa$?

Thanks ahead:)
 A: Yes. This follows immediately from the following lemma.

Lemma: If $X$ is ccc, every locally finite open cover of $X$ is countable.
Proof: Let $\mathscr{U}$ be a locally finite open cover of $X$. For each $x\in X$ let $V_x$ be an open nbhd of $x$ that meets only finitely many members of $\mathscr{U}$. For each $U\in\mathscr{U}$ pick any $x\in U$ and let $V_U=V_x$. Let $\mathscr{V}=\{V_U:U\in\mathscr{U}\}$.  Define an equivalence relation $\sim$ on $\mathscr{V}$ by $V\sim W$ iff there are $n\in\omega$ and $V_0=V,V_1,\ldots,V_n=W\in\mathscr{V}$ such that $V_k\cap V_{k+1}\ne\varnothing$ for $0\le k<n$. Each member of $\mathscr{V}$ meets only finitely many members of $\mathscr{U}$, so
each member of $\mathscr{V}$ meets only finitely many members of $\mathscr{V}$ as well, and each $\sim$-equivalence class is therefore countable. The union of each equivalence class is open, and these unions are pairwise disjoint, so there are only countably many equivalence classes. Thus, $\mathscr{V}$ is countable, and therefore $\mathscr{U}$ is countable. $\dashv$

