I would like to illustrate my lecture on p-adic numbers with some elementary results. I proved that the series $e^p=\sum_{n\ge0}\frac{p^n}{n!}$ converges in $\mathbb Q_p$ for every prime $p$.

Now I would like to teach that $e^p$ is irrational in $\mathbb Q_p$ by elementary methods (this is true by Mahler result on transcendence of $e^x$ in $p$-adic fields).

Do you know such an elementary proof. I can't find one in the literature...

Good catch! I meant $\frac{p^n}{n!}$ instead of $\frac1n$. I edited it.