A construction for the lower bound: take the matrix of rank $k-1$ with $2^{k-1}$ columns consisting of all $\{0,1\}$-combinations of the vectors $e_1+e_2, e_3+e_4 \dots, e_{2k-3}+e_{2k-2}$. Then add the column $e_1+e_3+e_5+\dots+e_{2k-1}$, which will increase the rank by $1$ and make all the rows distinct.

A construction for the lower bound: take the matrix of rank $k-1$ with $2^{k-1}$ columns consisting of all $\{0,1\}$-combinations of the vectors $e_1+e_2, e_3+e_4 \dots, e_{2k-3}+e_{2k-2}$. Then add the column $e_1+e_3+e_5+\dots+e_{2k-1}$, which will increase the rank by $1$ and make all the rows distinct.

Edit: If the rank was computed over $\mathbb{Z}_2$, the matrix with $2^k$ columns consisting of all $\{0,1\}$-combinations of the $k$ vectors above also has rank $k$ and no zero row or two identical rows:

\begin{pmatrix} 0&1&1&0\\ 0&1&0&1\\ 0&0&1&1 \end{pmatrix}