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 on arXiv).