We have the following theorem of Baker:

**Theorem 1.** Let $\alpha_1, \ldots, \alpha_m \in \mathbb{C}$ be algebraic numbers $\neq 0, 1$ such that $\log \alpha_1, \ldots, \log \alpha_m$ are linearly independent over $\mathbb{Q}$. Then $\log \alpha_1, \ldots, \log \alpha_m$ are linearly independent over the algebraic numbers.

Here I read that this theorem has an analog in the $p$-adic setting, so I guess it holds something like:

**Theorem 2 (?).** Let $p$ be a prime number and $\alpha_1, \ldots, \alpha_m \in \{x \in \mathbb{C}_p : |x - 1|_p < p^{-1/(p-1)}\}$ be algebraic numbers such that $\log_p \alpha_1, \ldots, \log_p \alpha_m$ (where $\log_p$ is the $p$-adic logarithm) are linearly independent over $\mathbb{Q}$. Then $\log_p \alpha_1, \ldots, \log_p \alpha_m$ are linearly independent over the algebraic numbers.

My question is: Is Theorem 2 true? Where I can find a reference?

Thanks.