For non-zero sequence $x=(x_1, x_2, \ldots)$ denote $L(x)=\min\{i:x_i\ne 0\}$ (leader of $x$). By Gauss elimination $M$ contains a basis $(p_1, \ldots, p_d) $ with distinct leaders $m_1<m_2<\ldots<m_d$ respectively. Let $f_1, f_2, \ldots$ denote consecutive standard basic vectors $e_k$ with $k\notin \{m_1,\ldots,m_d\}$. Consider the span $N$ of $M$ and $f_1, \ldots, f_s$ with very large $s$.On this space $N$, the coordinate functionals corresponding to $m_i$'s are uniformly (by $s$) bounded that is proved by induction. Now you cut all $m_j$'s on the corresponding level $d+s$ and get an $l^\infty$ space very close to $N$.