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Dress and Siebeneicher gave a significant generalization of the constructionof Witt vectors, by producing for any profinite group $G$, a ring-valuedfunctor $\mathbf{W}_G$. This paper gives a concrete interpretation of the rings$\mathbf{W}_{\mathbf{Z}_p^2}(k)$ where $k$ is a field of characteristic $p > 0$in terms of rings of Lipschitz continuous functions on the $p$-adic upper halfplane $\mathbf{P}^1(\mathbf{Q}_p)$. As a consequence we show that the Krulldimensions of the rings $\mathbf{W}_{\mathbf{Z}_p^d}(k)$ are infinite for $d\geq 2$ and we show the Teichm\"uller representatives form an analogue of thevan der Put basis for continuous functions on $\mathbf{Z}_p$.

Witt-Burnside functor attached to $\mathbf{Z}_p^2$ and $p$-adic Lipschitz continuous functions