Let $p$ be a prime number, and $K$ a finite extension of $\mathbb Q_p$ and $S=p^N\mathcal O_K$ where $\mathcal O_K$ the ring of integers of $K.$ I know that for $N>>0$ enough large the $p$-adic exponential $\exp_p$ converge on $S,$ so my question is : it is true that $\exp_p(S)\subset \mathcal O_K^*$?
Thank you for any help.