Let $M$ be an $n$-dimensional differentiable manifold $M$ and $X_1,\dots,X_k$ smooth vector fields on an open set $U$ of $M$ such that they are linearly independent in every point in $U$.
It is not difficult to show that for every $m\in U$ there are an open neighborhood $V\subset U$ of $m$ and smooth vector fields $X_{k+1},\dots,X_{n}$ defined in $V$ such that $\{X_1(p),\dots,X_n(p)\}$ is a basis of $T_pM$ for all $p\in V$.
Is is possible to find smooth vector fields $X_{k+1},\dots,X_n$ defined in the whole open set $U$ such that $\{X_1(p),\dots,X_n(p)\}$ is a basis of $T_pM$ for all $p\in U$?
I guess this is not true, but I cannot find any counterexample.