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=8TeV 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.