The answer is no. Let $P_n$ be the natural projection from $Z_{1p} :=\ell_1(\ell_p)$ onto the sum of the first $n$ copies of $\ell_p$. Let $Z$ be any subspace of $Z_{1p}$ that contains no isomorphic copy of $\ell_p$. Then the restriction of $P_n$ to $Z $ is strictly singular, so there is a norm one vector $x_n$ in $Z $ with $\|P_n z\|<1/n$. Do a standard gliding hump argument to deduce that $x_n$ has a subsequence equivalent to the unit vector basis of $\ell_1$.

The answer is no. Let $P_n$ be the natural projection from $Z_{1p} :=\ell_1(\ell_p)$ onto the sum of the first $n$ copies of $\ell_p$. Let $Z$ be any subspace of $Z_{1p}$ that contains no isomorphic copy of $\ell_p$. Then the restriction of $P_n$ to $Z $ is strictly singular, so there is a norm one vector $x_n$ in $Z $ with $\|P_n x_n\|<1/n$. Do a standard gliding hump argument to deduce that $x_n$ has a subsequence equivalent to the unit vector basis of $\ell_1$.