AbstractCUPID-0 is the first pilot experiment of CUPID, a next-generation project searching for neutrinoless 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 year to search for the neutrinoless double beta decay of $^{70}$70Zn and for the neutrinoless positron-emitting electron capture of $^{64}$64Zn. We found no evidence for these decays and set 90$\%$% credible interval limits of $\hbox {T}_{1/2}^{0ν β β }$T1/20νββ($^{70}$70Zn) > 1.6 $10^{21}$1021 year and $\hbox {T}_{1/2}^{0ν EC β +}$T1/20νECβ+($^{64}$64Zn) > 1.2$\times 10^{22}$×1022 year, surpassing by more than one order of magnitude the previous experimental results (Belli et al. in J Phys G 38(11):115107, https://doi.org/10.1088/0954-3899/38/11/115107, 2011).