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 $$\sqrt{s}=\mathrm {8~TeV}$$ 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.