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It is a well-known result of Kadec that every two separable infinite dimensional Banach spaces are homeomorphic. Also in large classes of nonseparable Banach spaces (perhaps all) the density character of a Banach space is its only topological invariant (see the book [2] for details). The situation changes considerably if we consider uniform homeomorphisms. Several results are known which prove the nonexistence of uniform homeomorphisms between certain Banach spaces of the same density character. As a matter of fact, the following problem was raised by many mathematicians: Do there exist two nonisomorphic Banach spaces which are uniformly homeomorphic? (Le. does the uniform structure of a Banach space determine its linear structure?) For a recent survey of results related to this problem see [3].While studying the question of existence of nonlinear liftings, we found in a surprisingly simple manner an example which answers this problem. Let T be a set of the cardinality of the continuum. Then c 0 (T) is lipschitz equivalent to a certain closed subspace X of l^ (i.e. there is a map T from c 0 (r) onto X so that T and T~x satisfy a lipschitz condition). Since there is no sequence of continuous linear functionals which separate the points in c 0 (T), this space is not isomorphic to a subspace of /«.Let UD F be Banach spaces and let admits a lipschitz (resp. uniformly continuous) lifting if there is a lipschitz (resp. uniformly continuous) map \p: UjV-• C/so that

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