In this letter we characterise IRAS12556-7731 as the first lithium-rich M-type giant. Based on its late spectral type and high lithium content, and because of its proximity in angular distance to the ChamaeleonII star-forming region, the star was misclassified as a young low-mass star in a previous work. Based on HARPS data, synthetic spectral modelling, and proper motions, we derive the astrophysical parameters and kinematics of the star and discuss its evolutionary status. This solar-mass red giant (Teff=3460+/-60K and log(g)=0.6+/-0.2) is characterised by a relatively fast rotation (v sin(i)~8km/s), slightly subsolar metallicity and a high-lithium abundance, A(Li)=2.4+/-0.2dex. We discuss IRAS12556-7731 within the context of other known lithium-rich K-type giants. Because it is close to the tip of the red giant branch, IRAS12556-7731 is the coolest lithium-rich giant known so far, and it is among the least massive and most luminous giants where enhancement of lithium has been detected. Among several possible explanations, we cannot preclude the possibility that the lithium enhancement and rapid rotation of the star were triggered by the engulfment of a brown dwarf or a planet.
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