Although unobservable in the standard model, charged lepton flavour violating (LVF) processes are predicted to be enhanced in new physics extensions. We present the final results of a search for electron-muon flavour violation in 𝛶(3𝑆)→ e^± μ^∓ decays using data collected with the BaBar detector at the SLAC PEP-II e^+e^− collider operating with a 10.36 GeV centre-of-mass energy. The search was conducted using a data sample of 118 million 𝛶(3S) mesons from 27 fb^−1 of data and is the first search for electron-muon LFV decays of a b quark and b antiquark bound state. No evidence for a signal is found, and we set a limit on the branching fraction of 𝛶(3S)→e^± μ^∓ and interpret it as a limit on the energy scale divided by the coupling-squared of relevant LFV new physics (NP):Λ_NP/(g_NP)^2 > 80 TeV.