The running of the top quark mass is experimentally investigated for the first time. The mass of the top quark in the modified minimal subtraction (MS‾) renormalization scheme is extracted from a comparison of the differential top quark-antiquark (tt¯) cross section as a function of the invariant mass of the tt¯ system to next-to-leading-order theoretical predictions. The differential cross section is determined at the parton level by means of a maximum-likelihood fit to distributions of final-state observables. The analysis is performed using tt¯ candidate events in the e± μ∓ channel in proton-proton collision data at a centre-of-mass energy of 13 TeV recorded by the CMS detector at the CERN LHC in 2016, corresponding to an integrated luminosity of 35.9fb−1. The extracted running is found to be compatible with the scale dependence predicted by the corresponding renormalization group equation. In this analysis, the running is probed up to a scale of the order of 1 TeV.