Continuity of differential operators for nonarchimedean Banach algebras

Given a nonarchimedean field $K$ and a commutative, noetherian, Banach $K$-algebra $A$, we study continuity of $K$-linear differential operators (in the sense of Grothendieck) between finitely generated Banach $A$-modules. When $K$ is of characteristic zero we show that every such operator is continuous if and only if $A/\mathfrak{m}$ is a finite extension of $K$ for every maximal ideal $\mathfrak{m}\subset A$.