A measure on a profinite group $\Gamma$ with values in a $p$-adic ring $\cal O$ and its reinterpretation as an element of the Iwasawa algebra $\Lambda_{\cal O}={\cal O}[[\Gamma]]$ are defined in textbooks under the hypothesis that $\cal O$ is the ring of integers in a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers.
I understand that the reason of this is so that the Iwasawa algebra $\Lambda_{\cal O}$ is compact, although I am not sure exactly at what point this becomes useful.
In particular, I see no reason why the theory shouldn't work for any $p$-adic ring $\cal O$. After all in the case $\Gamma=\mathbb{Z}_p$ Mahler's theory of interpolation of $R$-valued continuous functions works for any ring $R$.