Mapping Analytic sets onto cubes by little Lipschitz functions

A mapping $f:X\to Y$ between metric spaces is called \emph{little Lipschitz} if the quantity $$\operatorname{lip}(f(x)=\liminf_{r\to0}\frac{\operatorname{diam} f(B(x,r))}{r}$$ is finite for every $x\in X$. We prove that if a compact (or, more generally, analytic) metric space has packing dimension greater than $n$, then $X$ can be mapped onto an $n$-dimensional cube by a little Lipschitz function. The result requires two facts that are interesing in their own right. First, an analytic metric space $X$ contains, for any $\varepsilon>0$, a compact subset $S$ that embeds into an ultrametric space by a Lipschitz map, and $\dim_P S\geq\dim_P X-\varepsilon$. Second, a little Lipschitz function on a closed subset admits a little Lipschitz extension.