Let $X$ be a Banach space with a Schauder basis, and $Y$ a Banach space. Let $P_N$ denote the coordinate projections relative to the basis of $X$, and let $X_N$ denote their ranges. Specifically, $\{x_n\}_{n=1}^{\infty}$ is a basis in $X$ and $$P_N(x)=\sum_{n=1}^Na_nx_n,\quad\hbox{with}\quad x=\sum_{n=1}^{\infty}a_nx_n$$ Assume that for each $N$, the finite dimensional space $X_N$ is isometric to a subspace of $Y$. Does it follow that $X$ is isometric to a subspace of $Y$?

If $Y=L_p$ with $1\leq p\leq 2$ then the answer is affirmative, but the general case escapes me.