Results are presented for a test of the compositeness of the heaviest charged lepton, τ, using data collected by the CMS experiment in proton-proton collisions at a center-of-mass energy of 13 TeV at the CERN LHC. The data were collected in 2016–2018 and correspond to an integrated luminosity of 138 fb−1. This analysis searches for tau lepton pair production in which one of the tau leptons is produced in an excited state and decays to a ground state tau lepton and a photon. The event selection consists of two isolated tau lepton decay candidates and a high-energy photon. The mass of the excited tau lepton is reconstructed using the missing transverse momentum in the event, assuming the momentum of the neutrinos from each tau lepton decay are aligned with the visible decay products. No excess of events above the standard model background prediction is observed. This null result is used to set lower bounds on the excited tau lepton mass. For a compositeness scale Λ equal to the excited tau lepton mass, excited tau leptons with masses below 4700 GeV are excluded at 95% confidence level; for Λ = 10 TeV this exclusion is set at 2800 GeV. This is the first experimental result covering this production and decay process in the excited tau mass range above 175 GeV.