algebraic number theory – A Tate-Sen theorem mod $p$

Let $p$ be a prime number, $G=textrm{Gal}(overline{mathbb{Q}}_p/mathbb{Q}_p)$, and $chi:Grightarrowmathbb{Z}_p^times$ the cyclotomic character. Let $mathbb{C}_p$ denote the completion of the algebraic closure of $mathbb{Q}_p$ and $mathcal{O}_{mathbb{C}_p}$ denote its ring of integers.

The Tate-Sen theorem implies that (among many other things) $H^0(G,mathbb{C}_p(chi))=0$.

Question: Is $H^0(G,mathcal{O}_{mathbb{C}_p}(chi)otimes mathbb{F}_p)=0$?

If the answer is no, do the Galois invariants generate (over $mathcal{O}_{mathbb{C}_p}$), or are they killed by some power of $p$ less than 1?

Motivation: Basically, I am trying to understand if there is some remannt of Hodge-Tate theory modulo $p$.