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Abstract CUPID-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 $$^{70}$$ 70 Zn and for the neutrinoless positron-emitting electron capture of $$^{64}$$ 64 Zn. We found no evidence for these decays and set 90$$\%$$ % credible interval limits of $$\hbox {T}_{1/2}^{0\nu \beta \beta }$$ T1/20νββ ($$^{70}$$ 70 Zn) > 1.6 $$10^{21}$$ 1021 year and $$\hbox {T}_{1/2}^{0\nu EC \beta +}$$ T1/20νECβ+ ($$^{64}$$ 64 Zn) > 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).