continued from above....then if $(v_{i})_{i=1}^{\infty}$ were $1$-spreading to being with, it would be $1$-equivalent to its subsequence $v_{k_{1}},v_{k_{2}},\ldots$, in which case the sub-array $(x_{i}^{k_{j}})_{i=1,j\in\mathbb{N}}^{\infty}$ generates the same asymptotic model.

Thank you for your answer! To clarify: 1) the definition that I gave of an asymptotic model would be fine, provided that the $1$-spreading hypothesis is omitted. 2) As I understand it, we should extract a subsequence of $(v_{i})_{i=1}^{\infty}$ like this: $\lim_{i_{1}\to\infty}\|\lambda_{k_{1}}x_{i_{k}}^{k_{1}}\|= \lim_{i_{1}\to\infty}\left\Vert\sum_{k=1}^{k_{1}}\lambda_{k}x_{i_{k}}^{k}\right\Vert=\left\Vert\sum_{i=1}^{k_{1}}\lambda_{i}v_{i}\right\Vert_{V}=\|\lambda_{k_{1}}v_{k_{1}}\|_{V}$ where $k_{1}\leq i_{1}<i_{2}<\ldots<i_{k_{1}}$ and $\lambda_{k}=0$ for $k<k_{1}$

Thank you for your helpful answer!

Thanks for your helpful answer! As it happens, I am interested in Banach spaces that are known to admit $\ell_{1}$ as a unique spreading model and that (I believe) cannot contain $\ell_{p}^{n}$'s.

Thanks for your comment and reference suggestion! To clarify my statement, it is Corollary 3.4.6 and the following paragraph of this thesis.