  We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its
finiteness means that Lipschitz functions on an arbitrary subset of M can be
linearly extended to functions on M whose Lipschitz constants are enlarged by a
factor controlled by \lambda(M). We prove that \lambda(M) is finite for several
important classes of metric spaces. These include metric trees of arbitrary
cardinality, groups of polynomial growth, Gromov-hyperbolic groups, certain
classes of Riemannian manifolds of bounded geometry and finite direct sums of
arbitrary combinations of these objects. On the other hand we construct an
example of a two-dimensional Riemannian manifold M of bounded geometry for
which \lambda(M)=\infty.
