Let $(X,\tau)$ be a Hausdorff paracompact space. Let $\tau'$ be the smallest $P$-topology refining $(X,\tau)$, i.e. the topology which has for base the $G_\delta$-subsets of $(X,\tau)$.

Is it true that $(X,\tau')$ is again paracompact?

If not, does the result hold under extra assumptions (e.g. hereditary paracompactness or even more)?

**EDIT:** Given the first answer received, I'll add a question:

Is there a Hausdorff paracompact space $(X,\tau)$ where every point has a local base of size $\leq \omega_1$ such that $(X,\tau')$ is not paracompact?