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Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.

Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb Q$?

In the real case, it is known that it is true (see Duverney Théorie des nombres or Nishioka Mahler functions and transcendence) But I can not find any proof or reference for the $p$-adic case.

Thanks in advance for any hints or references for this problem.

Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.

Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb Q$?

In the complex case, it is known that it is true (see Duverney Théorie des nombres or Nishioka Mahler functions and transcendence) But I can not find any proof or reference for the $p$-adic case.

Thanks in advance for any hints or references for this problem.

Let $p$ a prime number and $a\in\overline{\mathbb Q_p}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.

Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb Q_p$?

In the real case, it is known that it is the case (see Duverney Théorie des nombres or Nishioka Mahler functions and transcendence) But I can not find any proof or reference for the $p$-adic case.

Thanks in advance for any hints or references for this problem.

Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.

Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb Q$?

In the real case, it is known that it is true (see Duverney Théorie des nombres or Nishioka Mahler functions and transcendence) But I can not find any proof or reference for the $p$-adic case.

Thanks in advance for any hints or references for this problem.

Let $p$ a prime number and $a\in\overline{\mathbb Q_p}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$. Does the number $\sum_{n\ge0}a^{2^n}$ is transcendental over $\mathbb Q_p$?

In the real case, it is knwon that it is the case (see Duverney Théorie des nombres or Nishioka Mahler functions and transcendence) But I can not find any proof or reference for the $p$-adic case.

Thanks in advance for any hints or references for this problem.

Let $p$ a prime number and $a\in\overline{\mathbb Q_p}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.

Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb Q_p$?

In the real case, it is known that it is the case (see Duverney Théorie des nombres or Nishioka Mahler functions and transcendence) But I can not find any proof or reference for the $p$-adic case.

Thanks in advance for any hints or references for this problem.