Containment of $c_0$ or $\ell_p$

Question

Suppose that $(x_n)$ is a sequence of unit vectors in a Banach space $X$ such that $$\mbox{dist}(x_m, S_{X_n})=1$$

for all $m > n$. Here $S_{X_n}$ stands for the unit sphere $\mbox{span}\{x_1, \ldots, x_n\}$. Can we conclude that $X$ contains an isomorphic copy of $c_0$ or $\ell_p$?

This question is motivated by the observation that the canonical bases of $c_0$ and $\ell_p$ have this property. I cannot think of any other example of a Banach space with this property.

Answer 1

No.

You can find such a sequence in any infinite dimensional normed space. See the proof of Lemma 1.4.22 in Megginson's An Introduction to Banach Space Theory. But, there are infinite dimensional Banach spaces that contain neither $c_0$ nor any $\ell_p$, such as Tsirelson's space.