We consider (\ell_1)-Rank-(r) Approximation over GF(2), where for a binary (m\times n) matrix ({\bf A}) and a positive integer (r), one seeks a binary matrix ({\bf B}) of rank at most (r), minimizing the column-sum norm (||{\bf A} -{\bf B}||_1). We show that for every (\epsilon\in (0, 1)), there is a randomized ((1+\epsilon))-approximation algorithm for (\ell_1)-Rank-(r) Approximation over GF(2) of running time (m^{O(1)}n^{O(2^{4r}\cdot \epsilon^{-4})}). This is the first polynomial time approximation scheme (PTAS) for this problem.