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].
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