Is $\sum_{n=0}^\infty n! \ne 0$ in every $\mathbb{Q}_p$?

Question

Is $\sum_{n=0}^\infty n! \ne 0$ in every $\mathbb{Q}_p$? I read that this was an open problem but the answer seemed obvious that it should be true since after the term p! every term added on will be small and not affect the previous digits to the right.