See paper for full list of authors - 36 pages in total, author list starting page 20, 3 figures, 2 tables, submitted to Physics Letters B, All figures including auxiliary figures are available at http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/STDM-2016-08/ ; Inclusive isolated-photon production in $pp$ collisions at a centre-of-mass energy of 13 TeV is studied with the ATLAS detector at the LHC using a data set with an integrated luminosity of 3.2 fb$^{-1}$. The cross section is measured as a function of the photon transverse energy above 125 GeV in different regions of photon pseudorapidity. Next-to-leading-order perturbative QCD and Monte Carlo event-generator predictions are compared to the cross-section measurements and provide an adequate description of the data.