Assume that $U \subseteq \mathbb{R}$ is open and $f_1,...,f_k : U \rightarrow \mathbb{R}^n$ are differentiable functions such that $(f_1,...,f_k)$ has rank $k$ everywhere.
Can we find more functions $f_{k+1},...,f_n : U \rightarrow \mathbb{R}^n$ such that $(f_1,...,f_n)$ has rank $n$ everywhere?
When $n$ is even we can take a matrix $A$ with no real eigenvalues; then $f_1$ and $f_2 := A \cdot f_1$ always form a rank-$2$ function. I don't know whether this can be extended to answer the question though.