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We find a counterexample to a conjecture of Ga{\l}\k{e}ski by constructingfor some positive integers $m<n$ a mapping $f\inC^1(\mathbb{R}^n,\mathbb{R}^n)$ satisfying $\mathrm{rank}\, Df\leq m$ that,even locally, cannot be uniformly approximated by $C^2$ mappings$f_\varepsilon$ satisfying the same rank constraint $\mathrm{rank}\,Df_\varepsilon\leq m$.

$C^1$ mappings in $\mathbb{R}^5$ with derivative of rank at most $3$  cannot be uniformly approximated by $C^2$ mappings with derivative of rank at  most 3