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 ($\mathrm{\overline{MS}}$) renormalization scheme is extracted from a comparison of the differential top quark-antiquark ($\mathrm{t\overline{t}}$) cross section as a function of the invariant mass of the $\mathrm{t\overline{t}}$ 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 $\mathrm{t\overline{t}}$ candidate events in the e$^\pm\mu^\mp$ 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 35.9 fb$^{-1}$. 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.