Let me expand my comment. I found the reference I had in mind: Ihara and Nakamura, Some examples for Anabelian geometry in high dimensions. The moduli space of $g$ dimensional principally polarized abelian varieties with level $n\ge 3$ structures is a $K(\pi, 1)$ because the universal cover, which is the Siegel upper half plane, is contractible. In particular, its fundamental group $\Gamma(n)\subset Sp_{2g}(\mathbb{Z})$ has finite cohomological dimension. However, when $g>1$, they show that profinite completion $\widehat{\Gamma(n)}$ has infinite cohomological dimension. So it cannot be a $K(\pi^{et},1)$ in the sense you gave.

Let me expand my comment. I found the reference I had in mind: Ihara and Nakamura, Some examples for Anabelian geometry in high dimensions. The moduli space of $g$ dimensional principally polarized abelian varieties with level $n\ge 3$ structures is a $K(\pi, 1)$ because the universal cover, which is the Siegel upper half plane, is contractible. In particular, its fundamental group $\Gamma(n)\subset Sp_{2g}(\mathbb{Z})$ has finite cohomological dimension. However, when $g>1$, they show that the profinite completion $\widehat{\Gamma(n)}$ has infinite cohomological dimension. So it cannot be a $K(\pi^{et},1)$ in the sense you gave.

Added To avoid any confusion, $\Gamma(n)\subset Sp_{2g}(\mathbb{Z})$ is the congruence subgroup of full level $n$, i.e. the kernel of the map to $Sp_{2g}(\mathbb{Z}/n)$. Since $n\ge 3$, this is known to act without fixed points on the Siegel upper half plane, so there is no need to worry about orbifolds above.