Cayley graphs are well known objects with interesting properties, also in the context of Moore graphs and digraphs. In $1978$ Bos\'ak extended Moore's property to the mixed setting (with arcs allowed along with edges), in the so called mixed Moore graphs: those having a unique trail--path? between pairs of vertices at a distance smaller than or equal to the diameter. In this paper we adapt Cayley's construction to mixed graphs and we show certain mixed Moore graphs are Cayley while some other cannot be Cayley.