Since the original question has been answered by Fedor Petrov and Robert Israel, here are more difficult questions (the terminology is preserved).
Let $m>n$. What is the smallest number $c(m,n)$ of colors needed to color the unit cube $I^m$ so that no color is $n$-distance connected. I need a rough (but meaningful) estimate from below or from above.
If $m=n+1$, is $c(m,n)=O(2^n/n)$ as in Fedor's answer?
What if $m=2n$?
Can $c(m,n)$ be bounded from above by a polynomial in $n$ for some/all $m\gg n$?
Update It looks like Fedor's proof gives the upper bound $c(m,n)\le O(2^n/n)$ for every $m > n$ (does not depend on $m$). The big question is whether it (or something similar) is also a lower bound or perhaps it is getting smaller with $m$.