TITLE: When does a pair of homotopic Lipschitz functions fail to admit a Lipschitz homotopy?
QUESTION [14 upvotes]: Let $(M,d)$ and $(N,\rho)$ be metric spaces. A function $f: M \to N$ is Lipschitz if there exists some constant $\kappa \geq 0$ so that $\rho(f(x),f(y))$ is smaller than $\kappa d(x,y)$ for all points $x,y \in M$.
If $f$ and $g$ are two Lipschitz maps from $M$ to $N$, a Lipschitz homotopy between them is a Lipschitz map $H: M \times [0,1] \to N$ for which $H(x,0) = f(x)$ and $H(x,1) = g(x)$ for each $x \in M$. Here we assume that the metric $\Delta$ on $M \times [0,1]$ is given by
$$\Delta\left((x,t),(y,s)\right) = \max\left\{d(x,y),|s-t|\right\}.$$
Here's the question which has been driving me nuts:

Is there an example of metric spaces $M, N$ and Lipschitz functions $f,g:M \to N$ which are homotopic, but not Lipschitz-homotopic?

If there is any justice in the world, the answer is an easy "yes". And in this case, I ask

Does the answer continue to be "yes" if $M$ and $N$ are assumed compact?

REPLY [2 votes]: I agree with Eremenko that 
there is no "justice" in the world of arbitrary metric spaces:-)
However, some justice is provided by the following beautiful result of 
Lang and Schlichenmaier, Theorem 1.5 in [1]. From this result one can conclude some results about Lipschitz homotopy of maps.

Theorem. Suppose that $X$ and $Y$ are metric spaces such that 
  the Nagata dimension of $X$ satisfies $\dim_N X\leq n$ and  $Y$ is Lipschitz $(n-1)$-connected. Then there
  is a constant $C\geq 1$ such that for any closed set $Z\subset X$ and
  any $L$-Lipschitz map $f:Z\to Y$ there is a $CL$-Lipschitz extension
  $F:X\to Y$.

The definitions of the Nagata dimension and Lipschitz $k$-connectivity are provided below:
Definition. A metric space $Y$ is Lipschitz $n$-connected for some integer
$n\geq 0$ if there is a constant $\gamma\geq 1$ such that for each 
$k\in \{0,1,\ldots,n\}$, every $L$-Lipschitz map 
$f:\mathbb{S}^k\to Y$ admits a $\gamma L$-Lipschitz extension 
$F:\mathbb{B}^{k+1}\to Y$.
Definition. The Nagata dimension $\dim_N X$ of a metric space $X$ is the least integer
$n$ with the property that there is $C>0$ such that for any $s>0$,
there is a covering $X=\bigcup_{i\in I} A_i$ such that

$\operatorname{diam} A_i\leq Cs$ for all $i\in I$;
Every ball $\mathbb{B}(x,s)$ intersects at most $n+1$ sets $A_i$.

If no such integer $n$ exists, then $\dim_N X=+\infty$.
The Nagata dimension can be regarded, in some sense, as a quantitative version of the topological dimension.
[1] U. Lang, T. Schlichenmaier,
Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. 
Int. Math. Res. Not. 2005, no. 58, 3625-3655.