$\sum\limits_{n=0}^{\infty}\dfrac{1}{n!}$ doesn't converge in $\mathbb{Q}_p$, however, $e^p:=\sum\limits_{n=0}^{\infty}\dfrac{p^n}{n!}$ does converge for $p\neq 2$. So my question is,

Are $e^p\in\mathbb{Q}_p$ for $p\neq2$ (and $e^4\in\mathbb{Q}_2$) known to be non-algebraic numbers?