N=4, d=1 Supersymmetric Hyper-Kaehler Sigma Models and Non-Abelian Monopole Background

We construct a Lagrangian formulation of \Nf supersymmetric mechanics with hyper-K\"{a}hler sigma models in a bosonic sector in the non-Abelian background gauge field. The resulting action includes a wide class of \Nf supersymmetric mechanics describing the motion of an isospin-carrying particle over spaces with non-trivial geometry. In two examples we discuss in details, the background fields are identified with the field of BPST instantons in the flat and Taub-NUT spaces.