Let $K$ be a complete nonarchimedean field. The classical Tate algebra $K\langle T \rangle$ has lots of automorphisms, e.g., any substitution $T\mapsto a_1T+a_2T^2+\cdots$, where $a_1\in \mathcal{O}_K^\times$, $a_n\in\mathfrak{m}_K$ for $n\geq 2$, and $a_n\to 0$ as $n\to \infty$.

This question is about perfectoid spaces. Let $K$ be a perfectoid field (in particular $K$ is a non-discrete valued field). Let $A$ be the ``perfectoid Tate algebra'' $K\langle T^{1/p^\infty}\rangle$. This is obtained by inverting $p$ on the $p$-adic completion of $\mathcal{O}_K[T^{1/p^\infty}]=\bigcup_{n\geq 1} \mathcal{O}_K[T^{1/p^n}]$.

If the characteristic of $K$ is $p$, then $K$ is perfect. In that case, the automorphisms of $K\langle T\rangle$ from the first paragraph extend uniquely to $A$. Other automorphisms arise by composing these with $p$th powers and roots.

My question is, are there any other automorphisms? For instance, suppose $K$ has characteristic $p$ and $\varpi\in K$ has positive valuation. Then the substitution $$ T\mapsto T+\varpi T^{1/p}+\varpi^2 T^{1/p^2}+\cdots $$ is certainly an endomorphism of $A$, but I cannot decide if it is an automorphism.