The inclusive WZ production cross section is measured in proton-proton collisions at a centre-of-mass energy of 13.6 TeV, using data collected during 2022 with the CMS detector, corresponding to an integrated luminosity of 34.7 fb−1. The measurement uses multileptonic final states and a simultaneous likelihood fit to the number of events in four different lepton flavour categories: eee, eeμ, μμe, and μμμ. The selection is optimized to minimize the number of background events, and relies on an efficient prompt lepton discrimination strategy. The WZ production cross section is measured in a phase space defined within a 30 GeV window around the Z boson mass, as σtotal (pp → WZ) = 55.2 ± 1.2 (stat) ± 1.2 (syst) ± 0.8 (lumi) ± 0.3 (theo) pb. In addition, the cross section is measured in a fiducial phase space closer to the detector-level requirements. All the measurements presented in this paper are in agreement with standard model predictions.