Extensions of Lipschitz maps into Hadamard spaces

We prove that every lambda -Lipschitz map f: S --> Y defined on a subset of an arbitrary metric space X possesses a c lambda -Lipschitz extension (f) over bar: X --> Y for some c = c(Y) greater than or equal to 1, provided Y is a Hadamard manifold which satisfies one of the following conditions: (i) Y has pinched negative sectional curvature, (ii) Y is homogeneous, (iii) Y is two-dimensional. In case (i) the constant c depends only on the dimension of Y and the pinching constant, in case (iii) one may take c := 4 root2. We obtain similar results for large classes of Hadamard spaces Y in the sense of Alexandrov.