= Solution
We use the standard characteristic-$p$ absolute Galois-group theorem, $\operatorname{cd}_p(K)\le1$, and the cohomological identification $\operatorname{Br}(K)=H^2(K,K_s^\times)$. Both are clearly stated inputs permitted for this question.
In characteristic $p$, the map $a\mapsto a^p$ on $K_s^\times$ is injective, although it need not be surjective: $p$th roots are purely inseparable. Thus it gives an exact sequence of discrete <Galois modules>
$$
1\longrightarrow K_s^\times\xrightarrow{\ p\ }K_s^\times
\longrightarrow Q\longrightarrow1,\qquad Q=K_s^\times/(K_s^\times)^p.
$$
The quotient $Q$ is killed by $p$. Since $\operatorname{cd}_p(K)\le1$, $H^2(K,Q)=0$. The long exact cohomology sequence contains
$$
\operatorname{Br}(K)\xrightarrow{\ p\ }\operatorname{Br}(K)\longrightarrow H^2(K,Q)=0.
$$
Therefore \b[multiplication by $p$ on $\operatorname{Br}(K)$ is surjective]. This is <Brauer p-divisibility in characteristic p>. The quotient-module sequence is the relevant one in characteristic $p$; the separable Kummer sequence with a nontrivial $\mu_p$ is not available there.
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