Is there a five-point metric space, where any 4-point subset of it can be isometrically embedded in the Euclidean plane $\mathbb{R}^2$, but the whole metric space can't, and all 10 distances are positive integers?
This question was inspired by this one where no restriction was put on the distances. This answer describes an example where three of the distances are equal, and are $\sqrt3$ as large as the other seven.
I wondered if it was possible with all distances integer.
What I've tried: Various planar sets of four points, e.g. a 4 by 3 rectangle; a rhombus 5, 5, 5, 5 with diagonals 8, 6; a trapezium 4, 2, 3, 2 with diagonals 4, 4; a parallelogram 7, 4, 7, 4 with diagonals 9, 7; and various subsets of the equiangular hexagon 5, 3, 5, 3, 5, 3 with $AD=BE=CF=8$ and other diagonals 7. I can't get any of them to work. It seems that those attempts fail because the distances have too many different values. But a planar set of four points whose distances have a mere two different values is no good for this problem, because the distances need to have a ratio of $1:\sqrt3$ (as in the cited example), $1:\sqrt2$ (square) or $1:\phi$ (regular pentagon).
