It is true if you make some mild assumption on the valuation ring $R$ and the Hopf algebra $H=R[G]$.
For example, let's assume that $H$ is free over $R$, and that $R$ is a discrete valuation ring.
The main ingredient in the proof in the paper you cited is the following fact about co-algebras over a field: if $C$ is a coalgebra over a field $K$, and $V\subseteq C$ is a finite dimensional subspace, then there exists a finite dimensional subcoalgebra $D\subseteq C$ such that $V\subseteq D$. The proof goes almost the same for discrete valuation rings as for fields. We need to assumption on $H$ to make sure that nothing too singular will occur.