The internal space of a cut and project scheme is required to be a locally compact Abelian group. There are not too much locally Abelian groups out there. Besides the reals, there are the fields of p-adic numbers $\mathbb{Q}_p$. Indeed, it turns out that some substitution tilings - like the chair and the sphinx - are model sets with p-adic internal spaces (In these two examples: $H=\mathbb{Q}_2 × \mathbb{Q}_2$). Others - like the Watanabe Ito Soma 8-fold tilings - are model sets with respect to products of Euclidean and p-adic internal spaces (here, $H=\mathbb{R}^2 × \mathbb{Q}_2$).