Let $\ell_1^n$ denote $\mathbb{C}^n$ endowed with the $\ell_1$-norm. Is it possible that a reflexive space contains isometric copies of all $\ell_1^n$s complemented by projections with norm bounded by some positive $\delta>0$. It seems unlikely but I don't know how to disprove it.
EDIT: Since the question has a trivial positive solution, I'd like to ask if the situation described above is possible in $\ell_p$-spaces for $p\in (1,\infty)$.