The Cryogenic Underground Observatory for Rare Events (CUORE) is a large-scale cryogenic experiment searching for neutrinoless double-beta decay (0νββ) in ¹³⁰Te. The CUORE detector is made of natural tellurium, providing the possibility of rare event searches on isotopes other than ¹³⁰Te. In this work we describe a search for neutrinoless positron-emitting electron capture (β+EC) decay in ¹²⁰Te with a total TeO₂ exposure of 355.7 kg yr, corresponding to 0.2405 kg yr of ¹²⁰Te. Albeit 0νββ with two final-state electrons represents the most promising channel, the emission of a positron and two 511-keV γ ’s make 0νβ+EC decay signature extremely clear. To fully exploit the potential offered by the detector modularity we include events with different topology and perform a simultaneous fit of five selected signal signatures. Using blinded data we extract a median exclusion sensitivity of 3.4 x 10²² yr at 90% credibility interval (C.I.). After unblinding we find no evidence of 0νβ+EC signal and set a 90% C.I. Bayesian lower limit of 2.9 x 10²² yr on the ¹²⁰Te half-life. This result improves by an order of magnitude the existing limit from the combined analysis of CUORE-0 and Cuoricino.