In Automorphisms of $\mathbb C_p$, K. Conrad showed that there exist uncountably many $\mathbb Q_p$-automorphisms of $\mathbb C_p$. I have a quite similar question.

Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be sequences in $\mathbb Q_p$. Does there exist an uncountable set $S$ of continuous automorphisms of $\mathbb C_p$ that leave invariant $\{b_n\mid n\in\mathbb N\}$ and such that the set of images of the sequence $(a_n)_{n\in\mathbb N}$ by the elements of $S$ is uncountable. I assume that for all $n\in\mathbb N$, $b_n\notin\mathbb Q(\smash{a_j}_{\mid j\in\mathbb N})$.