  We show that sets avoiding 6-term arithmetic progressions in $\mathbb{Z}_6^n$
have size at most $5.709^n$. It is also pointed out that the "product
construction" does not work in this setting, specially, we show that for the
extremal sizes in small dimensions we have $r_6(\mathbb{Z}_6)=5$,
$r_6(\mathbb{Z}_6^2)=25$ and $ 116\leq r_6(\mathbb{Z}_6^3)\leq 124$.
