  Given a split $\mathbb{P}$-functor $F:\mathcal{D}^b(X) \to \mathcal{D}^b(Y)$
between smooth projective varieties, we provide necessary and sufficient
conditions, in terms of the Hochschild cohomology of $X$, for it to become
spherical on the total space of a deformation of $Y$, and explain how the
spherical twist becomes the $\mathbb{P}$-twist on the special fibre. These
results generalise the object case, that is when $X$ is a point, which was
studied previously by Huybrechts and Thomas, and we show how they apply to the
$\mathbb{P}$-functor associated to the Hilbert scheme of points on a K3
surface. In the appendix we review and reorganise some technical results due to
Toda, relating to the interaction of Atiyah classes, the HKR-isomorphism, and
the characteristic morphism.
