A substitution rule with inflation factor $\sqrt2$
and two prototiles.
This makes it a candidate for being a model set with p-adic internal space.
This has not been checked so far, but in principle it is possible by the methods in [LMS03] or [FS].
[LMS03]
Lee, J E S and Moody, R V and Solomyak, B
Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
Discrete and Computational Geometry
2003,
29,
pp. 525-560,
MR1702375
[FS]
Frettlöh, D and Sing, B
Computing modular coincidences
Preprint