  It was conjectured that if $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ satisfies
$\operatorname{rank} Df\leq m<n$ everywhere in $\mathbb{R}^n$, then $f$ can be
uniformly approximated by $C^\infty$-mappings $g$ satisfying
$\operatorname{rank} Dg\leq m$ everywhere. While in general, there are
counterexamples to this conjecture, we prove that the answer is in the positive
when $m=1$. More precisely, if $m=1$, our result yields an almost-uniform
approximation of locally Lipschitz mappings $f:\Omega\to\mathbb{R}^n$,
satisfying $\operatorname{rank} Df\leq 1$ a.e., by $C^\infty$-mappings $g$ with
$\operatorname{rank} Dg\leq 1$, provided $\Omega\subset\mathbb{R}^n$ is simply
connected. The construction of the approximation employs techniques of analysis
on metric spaces, including the theory of metric trees ($\mathbb{R}$-trees).
