Summary: Frölicher spaces form a cartesian closed category which contains the category of smooth manifolds as a full subcategory. Therefore, mapping groups such as \(C^{\infty }(M,G)\) or \(\mathrm{Diff}(M)\), and also projective limits of Lie groups, are in a natural way objects of that category, and group operations are morphisms in the category. We call groups with this property Frölicher groups. One can define tangent spaces to Frölicher spaces, and in the present article we prove that, under a certain additional assumption, the tangent space at the identity of a Frölicher group can be equipped with a Lie bracket. We discuss an example which satisfies the additional assumption.