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Let $A$ be a strictly henselian discrete valuation ring. What is $\pi_1(\operatorname{Spec}(A) \setminus \{s\})$?

I thought it is a semidirect product of a pro-$p$-group (the wild ramification group) and \hat{\mathbf{Z}}'(1) (the tame fundamental group), but it is claimed that it is only the tame fundamental group.

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Presumably $A$ could be $W(\overline{\mathbb F}_p),$ the Witt ring of the algebraic closure of $\mathbb F_p$. Then $\pi_1(Spec(A)\setminus \{s\})$ is the Galois group of the completion of the maximal unramified extension of $\mathbb Q_p$, which, as you say, is an extension of tame inertia by wild inertia.

When you write "it is claimed that ...", what reference are you reading?