Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$?

There are some results of the type: each finite ultrametrics admits an isomeric embedding into any infinite dimensional Banach space. See Shkarin, S. A. Isometric embedding of finite ultrametric spaces in Banach spaces. Topology Appl. 142 (2004), no. 1-3, 13-17 for this, and also the papers: Dekster, B. V. Simplexes with prescribed edge lengths in Minkowski and Banach spaces. Acta Math. Hungar. 86 (2000), no. 4, 343-358; Faver, Timothy; Kochalski, Katelynn; Murugan, Mathav Kishore; Verheggen, Heidi; Wesson, Elizabeth; Weston, Anthony Roundness properties of ultrametric spaces. Glasg. Math. J. 56 (2014), no. 3, 519-535.

There is a related question on MathOverflow: Almost isometric embeddability implies isometric embeddability

Added on 9/29/2016: Related results were obtained by James Kilbane in https://arxiv.org/abs/1609.08971

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