  We investigate the Whyburn and weakly Whyburn property in the class of
$P$-spaces, that is spaces where every countable intersection of open sets is
open. We construct examples of non-weakly Whyburn $P$-spaces of size continuum,
thus giving a negative answer under CH to a question of Pelant, Tkachenko,
Tkachuk and Wilson. In addition, we show that the weak Kurepa Hypothesis (a
set-theoretic assumption weaker than CH) implies the existence of a non-weakly
Whyburn $P$-space of size $\aleph_2$. Finally, we consider the behavior of the
above-mentioned properties under products; we show in particular that the
product of a Lindel\"of weakly Whyburn P-space and a Lindel\"of Whyburn
$P$-space is weakly Whyburn, and we give a consistent example of a non-Whyburn
product of two Lindel\"of Whyburn $P$-spaces.
