I know that in a characteristic $0$ field, every irreducible polynomial is separable. This follows directly from the equivalence:
$$f \:\text{ separable} \iff \gcd(f,f') = 1$$
Now I'm reading Neukirch's Algebraic Number Theory and on page 143 (Henselian Fields) he says: Let $(K,v)$ be a nonarchimedean valued field, ($\hat K$, $\hat v$) its completion, ($K_v$,$v_v$) its separable closure in $\hat K$. When $K_v$ is algebraically closed in $\hat K$ then $\operatorname{char}(K)=0$ and Hensel's lemma holds in $\mathcal{O}_v$ (the valuation ring of $K_v$).
For me, it seems like he says that if every irreducible polynomial is separable, then $\operatorname{char}(K) = 0$ because the above means: Let $f$ be an irreducible polynomial with coefficients in $K$, then if $K_v$ is algebraically closed $f$ splits into linear factors over $K_v$ and since $K_v$ is a separable extension of $K$, $f$ is separable.
But is this true? And why does Hensel's lemma hold in $\mathcal{O}_v$?