 Solution

At a point where $df$ is surjective, choose charts and reorder the domain coordinates so that an $\ell \times \ell$ minor of the derivative is invertible. The map $F=(f_1,\dots ,f_{\ell },x_{\ell +1},\dots ,x_k)$ has invertible derivative. By the inverse function theorem it gives local coordinates, in which $f$ is the projection $(u_1,\dots ,u_k)\mapsto (u_1,\dots ,u_{\ell })$. This is the submersion theorem.