pℤₚ has index p in ℤₚ

[YaelDillies]

The index of $p\mathbb Z_p$ inside $\mathbb Z_p$ is $p$. The precise version I have stated is that the $\mathbb Z_p$-span of $p$ times any set $s$ of $p$-adics has index $p$ relative to the $\mathbb Z_p$-span of $s$.