  First we compute the $\mbox{S}^2$ partition function of the supersymmetric
$\mathbb{CP}^{N-1}$ model via localization and as a check we show that the
chiral ring structure can be correctly reproduced. For the $\mathbb{CP}^1$ case
we provide a concrete realisation of this ring in terms of Bessel functions. We
consider a weak coupling expansion in each topological sector and write it as a
finite number of perturbative corrections plus an infinite series of
instanton-anti-instanton contributions. To be able to apply resurgent analysis
we then consider a non-supersymmetric deformation of the localized model by
introducing a small unbalance between the number of bosons and fermions. The
perturbative expansion of the deformed model becomes asymptotic and we analyse
it within the framework of resurgence theory. Although the perturbative series
truncates when we send the deformation parameter to zero we can still
reconstruct non-perturbative physics out of the perturbative data in a nice
example of Cheshire cat resurgence in quantum field theory. We also show that
the same type of resurgence takes place when we consider an analytic
continuation in the number of chiral fields from $N$ to $r\in\mathbb{R}$.
Although for generic real $r$ supersymmetry is still formally preserved, we
find that the perturbative expansion of the supersymmetric partition function
becomes asymptotic so that we can use resurgent analysis and only at the end
take the limit of integer $r$ to recover the undeformed model.
