Let $A$ be a discrete valuation ring and $\pi$ a uniformizer.
Over $A$ we consider the Tate-algebra $$A\langle t \rangle =\{ f=\sum_{n=0}^\infty a_nt^n \mid a_n\in A, \lim_{n\to \infty} \lvert a_n\rvert =0 \}$$ inside the formal power series ring $A[[t]]$, and a finitely generated $A\langle t\rangle$-algebra $R$ inside $A[[t]]$, i.e. $$R:=A\langle t \rangle [f_1,\dots, f_m]$$ for some $f_1,\dots, f_m\in A[[t]].$
Question: Is the saturation of $R$ in $A[[t]]$ still finitely generated over $A\langle t \rangle$, i.e. the subalgebra $$R^{\text{sat}}:=R[\frac{1}{\pi}]\cap A[[t]]=\{ f \in A[[t]] \mid \pi^nf\in R \text{ for some }n \}? $$
Remark: If needed we can assume $A$ to be complete. The case that I am mostly interested in is that of $A=\mathbb{Z}_p$.
