It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p) = 0$ for any extension $L/k$ of degree divisible by $p^\infty$.
I am wondering if this result generalizes to positive characteristic. More concretely, I am interested in the field $F= \overline{\mathbb{F}}_p(s,t)$ (with $\overline{\mathbb{F}}_p$ the algebraic closure of $\mathbb{F}_p$ and $s,t$ transcendental over $\mathbb{F}_p$), which is of cohomological dimension $\leq 2$ in analogy with $k$ above. My question is whether the Brauer group also trivializes as above, i.e., passing to the radical closure $F^{\text{rad}}$ of $F$, does $\text{Br}(F^{\text{rad}}) = 0$?
