A search for the rare 𝜂 →𝜇+⁢𝜇−⁢𝜇+⁢𝜇− double-Dalitz decay is performed using a sample of proton-proton collisions, collected by the CMS experiment at the CERN LHC with high-rate muon triggers during 2017 and 2018 and corresponding to an integrated luminosity of 101  fb−1. A signal having a statistical significance well in excess of 5 standard deviations is observed. Using the 𝜂→𝜇+⁢𝜇− decay as normalization, the branching fraction ℬ⁡(𝜂→𝜇+⁢𝜇−⁢𝜇+⁢𝜇−)=[5.0±0.8⁢(stat)±0.7⁢(syst)±0.7⁢(ℬ2⁢𝜇)]×10−9 is measured, where the last term is the uncertainty in the normalization channel branching fraction. This work achieves an improved precision of over 5 orders of magnitude compared to previous results, leading to the first measurement of this branching fraction, which is found to agree with theoretical predictions.