Does a family of linearly independent injective maps have a vector with linearly independent images?

This fails for $n=3.$ Consider $f_i$ with matrix representations

$$ \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}, \begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}, \begin{pmatrix}1&0&1\\0&1&0\\0&0&1\end{pmatrix}. $$

Or in other words: $\mathrm{id}, \mathrm{id}+e_1e_2^*, \mathrm{id}+e_1e_3^*.$ Then for any $v,$ all three vectors $f_i(v)$ lies in the space spanned by $\{v,e_1\}.$