Mapping analytic sets onto cubes by little Lipschitz functions

A mapping f : X -> Y between metric spaces is called little Lipschitz if the lip f(x) = lim inf (r -> 0) diam f(B(x, r)/r is finite for every x is an element of X. We prove that if a compact ( or, more generally, analytic) metric space has packing dimension greater than n, then it can be mapped onto an n- dimensional cube by a little Lipschitz function. The result requires two facts that are interesting in their own right. First, an analytic metric space X contains, for any epsilon > 0, a compact subset S that embeds into an ultrametric space by a Lipschitz map, and dimP S >= dimP X - epsilon. Second, a little Lipschitz function on a closed subset admits a little Lipschitz extension.