Measurements of fiducial integrated and differential cross sections for inclusive $W+$, $W-$ and $Z$ boson production are reported. They are based on $25.0\pm 0.5$ pb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $pp$ collision data at $s=5.02$ TeV collected with the ATLAS detector at the CERN Large Hadron Collider. Electron and muon decay channels are analysed, and the combined $W+$, $W-$ and $Z$ integrated cross sections are found to be $\sigma W+$ = 2266 $\pm$ 9 (stat) $\pm$ 29 (syst) $\pm$ 43 (lumi) pb, $\sigma W-$ = 1401 $\pm$ 7 (stat) $\pm$ 18 (syst) $\pm$ 27 (lumi) pb, and $\sigma Z$ = 374.5 $\pm$ 3.4 (stat) $\pm$ 3.6 (syst) $\pm$ 7.0 (lumi) pb, in good agreement with next-to-next-to-leading-order QCD cross-section calculations. These measurements serve as references for Pb+Pb interactions at the LHC at this nucleon--nucleon centre-of-mass energy.