Short answer
At present nobody knows the answer; no concrete pair
  F ⊂ ℓ₂ (finite) and X (infinite–dimensional Banach)
with “F does not embed isometrically into X’’ has been produced.
Equally, nobody has been able to prove that such a counter-example does not exist. The problem is therefore still open.

What is known

1. Positive results (existence of many embeddings).
   • Any finite metric space embeds isometrically in some ℓ∞ⁿ, hence every Banach space that contains arbitrarily large finite-dimensional ℓ∞-spaces (for example the spaces ℓ∞, L∞, C(K) with K uncountable, etc.) already contains an isometric copy of every finite subset of every metric space, in particular of every finite subset of ℓ₂.
   • Shkarin (2004) – every finite ultrametric embeds isometrically in every infinite-dimensional Banach space.
   • Dekster (2000) – finite Euclidean simplices with prescribed edge lengths embed in every infinite-dimensional Banach space.
   • Faver et al. (2014) – the same is true for all finite metrics that are “roundness-2” ultrametrics.
   These results show that very large families of finite subsets of ℓ₂ always embed, so a possible counter-example F would have to be quite special.

2. Partial negative information (what a counter-example would have to look like). All of the following is due to J. Kilbane (arXiv 1609.08971, arXiv 1704.00319).

   (i) If X is an infinite-dimensional Banach space that does not contain an isometric copy of every finite subset of ℓ₂, then X can be chosen to be separable and to contain no subspace isomorphic to ℓ₂. In fact, for every such X there is an n≤8 and an n-point subset F⊂ℓ₂⁴ that does not embed isometrically into X.

   (ii) The class
   𝔽 := {F ⊂ ℓ₂ finite : there is some infinite-dimensional X with F↛X isometrically}
   (if non-empty) is “small’’: up to isometry it has cardinality 2^{ℵ₀}; more precisely, for each fixed n the set of n-point counter-examples is meagre in the Polish space of all n-point metric spaces of negative type.

   These results restrict quite drastically where a counter-example could hide, but they do not actually produce such an example.

3. Very small configurations.
   For 2- and 3-point subsets of ℓ₂ it is easy to embed them isometrically into every Banach space (one and two distances can always be realised). No example with 4, 5, 6, 7 or 8 points is presently known.

4. Relation to other open problems.
   The question is a metric version of the classical linear problem “Does every infinite-dimensional Banach space contain a 2-dimensional Hilbert subspace?’’ – known to have a negative answer linearly (consider c₀), but still unresolved metrically. It also interacts with longstanding questions on equilateral sets, on the Ribe programme, and on metric characterisations of uniform convexity/smoothness.

To summarise • Nobody has produced a finite set of points in ℓ₂ that fails to sit isometrically in some infinite-dimensional Banach space,
• nor has anybody proved that such a set cannot exist. Hence the problem remains open, although extensive partial work (noted above) shows that any counter-example would have to be quite special and rather small.