Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?

It seems to me that it is an interesting question. It can be regarded as a special case of a recent question Under which conditions is it possible to find points with same distances under bi-Lipschitz map (which was closed) and is related to my (unanswered) question Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

In connection with this question it is worth mentioning that there is a theory developed by Krivine in Ann. Math. (2) 104, 1-29 (1976) (with important additions by Maurey and Pisier (Stud. Math. 58, 45-90 (1976)) and further simplifications by other authors, see Chapter 12 in the book by Benyamini-Lindenstrauss on Geometric Nonlinear Functional Analysis or Part II in the book Milman-Schechtman on Asymptotic Theory) which implies that for any $\varepsilon>0$ any finite-dimensional subspace of $\ell_p$ embeds into any Banach space isomorphic to $\ell_p$ with distortion $\le (1+\varepsilon)$.

Added on 4/3/2017: In a recent paper James Kilbane proved that the set of possible counterexamples (if they exist) is small in a certain sense.