Let $p$ be a prime and denote by $\mathbb{Z}_p^{\times}$ the group of $p$-adic units. Suppose that $\chi$ is a character $\chi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}^{\times}$. Then it is well known that

$\mathbb{Z}_p^{\times} \simeq (\mathbb{Z}/ p \mathbb{Z})^{\times} \times (1+ p \mathbb{Z}_p) $.

Do we have that the kernel of $\chi$ contains $(1+ p \mathbb{Z}_p)$ ? Any help or reference would be appreciated.