The goal of this question is to develop further the discussion initiated in Under which conditions is it possible to find points with same distances under bi-Lipschitz map. The mentioned question was closed because the author stated the question somewhat vaguely, but I think that the corresponding direction deserves attention.

**Definition:** We say that a metric space $X$ is *finitely
isometrically persistent* if for any metric space $Y$ which is
bilipschitz equivalent to $X$ (this means that there exists a
bijective Lipschitz map $F:X\to Y$ such that $F^{-1}$ is also
Lipschitz) and any finite set $A\subset X$ there is a subset
$B\subset Y$ such that $A$ and $B$ (with the induced metrics) are
isometric.

It is easy to find numerous examples of spaces which are not finitely isometrically persistent.

**Question:** Does there exist an infinite separable metric space which is
finitely isometrically persistent?

Restricting the category of metric spaces and (possibly also) the category of allowed maps $F:X\to Y$ we get many questions, some of which are definitely interesting. Some of them were asked in the question mentioned above, Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?, and Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces.

One can also consider suitably defined $\varepsilon$-*isometric
finite persistence*. Many of the interesting and important results
in the theory of Banach spaces and theory of metric embeddings can
be stated in these terms. An interesting counterexample (trees,
Theorem 1.12 in Mendel-Naor, JEMS, 15 (2013), 287-337) is also
known.

**Inserted after the question was answered:** The answer of Nik Weaver (see below) seems to show that in the study of this question we have to restrict ourselves, for example, to Banach spaces and linear isomorphisms (one can try something in between this and the general metric category).