Let $K$ be a non-archimedean field of characteristic zero and $||.||:K\to \mathbb{R}_{\geq 0}$ be its absolute value.
Define the Washnitzer Algebra as: $$W_n=\{\sum_{u\in \mathbb{Z}_{\geq 0}^n} \in K[[X]]: \text{ there exists some } \rho>1 \text{ such that } ||a_u||\rho^{|u|}\to 0 \text{ as } |u|\to \infty\}$$ where $|u|=u_1+u_2+\ldots+u_n$ for $u=(u_1,\ldots,u_n)$ and $\rho\in\mathbb{R}$.
Now, set $n=1$, so $$W_1=\{\sum_{n\in\mathbb{Z_{\geq 0}}}\in K[[X]]: \text{ there exists some } \rho>1 \text{ such that } ||a_n||\rho^{n}\to 0 \text{ as } n\to \infty\} $$
The question is the derivation map on $W_1$ defined as $$\partial: W_1\to W_1, \qquad \sum_{n\in\mathbb{Z_{\geq 0}}} a_n X^n \to \sum_{n\in\mathbb{Z_{> 0}}} na_nX^{n-1}$$ is a surjection or not?
Essentially, one needs to show that if $\sum_{n\in\mathbb{Z_{\geq 0}}}a_nX^n$ is in $W_1$ then we also have $\sum_{n\in\mathbb{Z_{> 0}}} \frac{a_{n-1}}{n}X^n $ (which is the formal integration of the initial series) is also in $W_1$.
More spesifically, if there exists some $\rho>1$ such that $$\lim_{n\to\infty} ||a_n|| \rho^n=0$$ then is there a $\rho_1$ such that $$\lim_{n\to\infty} ||{\frac{a_{n-1}}{n}}||{\rho_1}^n=0$$ ??