We investigate the Whyburn and weakly Whyburn property in the class of
$P$-spaces, that is spaces where every $G_\delta$ set 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.