Why is $O_{\mathbb{C}_p}/p$ not perfect?

Question

I was reading the beautiful Brinon and Conrad's introduction to $p$-adic Hodge theory (link) and I came to the introduction of de rham period ring. During the book, they always mention that the quotient of the ring of integers of the completion of the algebraic closure of $\mathbb{Q}_p$ modulo the ideal generated by $p$ is not perfect. How to prove it? Thank you for any suggestion.

Answer 1

Being perfect would mean that the map $x\mapsto x^p$ is bijective, or in other words, every element would have a unique $p$-th root and unique $p$-th power. Raising to the $p$-th power also raises the (multiplicative) valuation $|x|$ of an element $x$ to the $p$-th power. Now with $0 < c := |p| < 1$, can you see a whole bunch of elements in $O_{\mathbb{C}_p}$ which are nontrivial mod $p$, but whose $p$-th powers become $0$ mod $p$?