How to prove that $\mathbb{C}_p$ is not locally compact?

Question

Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of an algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. How can one show that $\mathbb{C}_p$ is not locally compact? I know that by scaling and translation, $\mathbb{C}_p$ is locally compact if and only if the closed unit disc $\overline{D}(1):=\{x\in\mathbb{C}_p\ |\ |x|\leq 1\}$ is compact. So in order to prove that $\mathbb{C}_p$ is not locally compact, one has to show that $\overline{D}(1)$ is not compact, which by completeness amounts to showing that $\overline{D}(1)$ is not totally bounded. But how would one go about proving this?