  CUPID-0 is the first pilot experiment of CUPID, a next-generation project
searching for neutrino-less double beta decay. In its first scientific run,
CUPID-0 operated 26 ZnSe cryogenic calorimeters coupled to light detectors in
the underground Laboratori Nazionali del Gran Sasso. In this work, we analyzed
a ZnSe exposure of 11.34 kg$\times$yr to search for the neutrino-less double
beta decay of $^{70}$Zn and for the neutrino-less positron-emitting electron
capture of $^{64}$Zn. We found no evidence for these decays and set 90$\%$
credible interval limits of ${\rm T}_{1/2}^{0\nu\beta\beta}(^{70}{\rm Zn}) >
1.6 \times 10^{21}$ yr and ${\rm T}_{1/2}^{0\nu EC \beta+}(^{64}{\rm Zn}) > 1.2
\times 10^{22}$ yr, surpassing by almost two orders of magnitude the previous
experimental results
