**There is a counterexample.** 

**Example.** There is $f\in C^1(\mathbb{R}^5,\mathbb{R}^5)$ with ${\rm rank}\, Df\leq 3$ that cannot be approximated in the supremum norm by mappings 
$g\in C^2(\mathbb{R}^5,\mathbb{R}^5)$ satisfying ${\rm rank}\, Dg\leq 3$.

**Example.** *There is $f\in C^1(\mathbb{R}^7,\mathbb{R}^7)$, ${\rm rank}\, Df\leq 4$, that cannot be approximated in the supremum norm by mappings
$g\in C^3(\mathbb{R}^7,\mathbb{R}^7)$ satisfying ${\rm rank}\, Dg\leq 4$.*

Both examples are special cases of the following result proved in [1]:


> **Theorem.** *Suppose that $m+1\leq k<2m-1$, $\ell\geq k+1$, $r\geq m+1$, and the homotopy group $\pi_k(S^m)$ is non-trivial.  Then there
> is a map $f\in C^1(\mathbb{R}^\ell, \mathbb{R}^r)$ with ${\rm rank}\,
 Df\leq m $ in $\mathbb{R}^\ell$ that cannot be approximated by maps of
> class $ C^{k-m+1}$.*

You can find infinitely many more examples.

[1] **P. Goldstein, P. Hajlasz,** $C^1$ mappings in $\mathbb{R}^5$ with 
${\rm rank}\, Df\leq 3$ cannot be
uniformly approximated by $C^2$ mappings with ${\rm rank}\, Df\leq 3$. Preprint.

The paper will be available on arXiv soon and then I will add a link. The main result is a counterexample to a conjecture of Galeski. Is that you?