We present the first measurement of ZZ$\gamma$ production in the final state with four charged leptons and a photon using the full LHC Run 2 data set. This analysis uses the proton-proton collision data at a center-of-mass energy of $13\,\mathrm{TeV}$ collected by the CMS detector from 2016 to 2018 with a total integrated luminosity of $138\,\mathrm{fb^{-1}}$. The first evidence for the triboson process $\mathrm{pp} \rightarrow ZZ\gamma \rightarrow 4\ell \gamma$ is reported, with the observed (expected) significance found to be 3.7 (3.0) standard deviations. The corresponding measured (expected) fiducial cross section is $ 56 {}^{+34}_{-28}\,\mathrm{ab}$ ($48 \pm 23\,\mathrm{ab}$). Furthermore, an additional measurement is performed by including the initial and final state radiation component in the signal definition. In this case, an observed (expected) significance of 5.1 (4.1) standard deviations is found for the electroweak production of $\mathrm{pp} \rightarrow 4\ell \gamma$, and the corresponding measured (expected) fiducial cross section of $162 {}^{+66}_{-58}\,\mathrm{ab}$ ($100 \pm 31\,\mathrm{ab}$). Both measurements are in agreement with the Standard Model predictions at the level of one standard deviation.