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A recent result of Freeman, Odell, Sari, and Zheng states that whenever aseparable Banach space not containing $\ell_1$ has the property that allasymptotic models generated by weakly null sequences are equivalent to the unitvector basis of $c_0$ then the space is Asymptotic $c_0$. We show that if wereplace $c_0$ with $\ell_1$ then this result is no longer true. Moreover, astronger result of B. Maurey - H. P. Rosenthal type is presented, namely, thereexists a reflexive Banach space with an unconditional basis admitting $\ell_1$as a unique asymptotic model whereas any subsequence of the basis generates anon-Asymptotic $\ell_1$ subspace.

Non-asymptotic $\ell_1$ spaces with unique $\ell_1$ asymptotic model