A search for the production of single-top-quarks in association with missing energy is performed in proton–proton collisions at a centre-of-mass energy of s√=8 TeV with the ATLAS experiment at the large hadron collider using data collected in 2012, corresponding to an integrated luminosity of 20.3 fb−1. In this search, the W boson from the top quark is required to decay into an electron or a muon and a neutrino. No deviation from the standard model prediction is observed, and upper limits are set on the production cross-section for resonant and non-resonant production of an invisible exotic state in association with a right-handed top quark. In the case of resonant production, for a spin-0 resonance with a mass of 500 GeV, an effective coupling strength above 0.15 is excluded at 95% confidence level for the top quark and an invisible spin-1/2 state with mass between 0 and 100 GeV. In the case of non-resonant production, an effective coupling strength above 0.2 is excluded at 95% confidence level for the top quark and an invisible spin-1 state with mass between 0 and 657 GeV.