Finite field extension of $\mathbb{Z}_p$

Question

Let $K$ be a finite extension of $\mathbb{Z}_p$ and let $P$ be the unique maximal ideal of $K$. We have $\mathbb{Z}_p\subseteq\mathcal{O}_K$ and that $P\cap\mathbb{Z}_p=p\mathbb{Z}_p$. The claim is that $\mathcal{O}_K$ is an extension of $\mathbb{F}_p=\mathbb{Z}_p/p\mathbb{Z}_p$. Why is this true? If we let $x+p\mathbb{Z}_p\in\mathbb{F}_p$ I'm not sure why this implies the result.