Suppose $\sigma(t)=t$, then, after applying an automorphism to $R = \mathbb{F}_3[[T]]$, we may assume $t=T^k$ for some $k>0$. So then $\sigma(T)=\zeta T$ for some $k$-th root of unity $\zeta$, hence $\sigma$ has finite order. Conversely, if $\sigma$ has finite order $k$, then symmetric expressions in $T$, $\sigma(T)$, $\ldots$, $\sigma^{(k-1)}(T)$ should give plenty of power series on which $\sigma$ acts trivially.