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We study the set of $p$-adic Gibbs measures of the $q$-states Potts model onthe Cayley tree of order three. We prove the vastness of the periodic $p$-adicGibbs measures for such model by showing the chaotic behavior of thecorrespondence Potts--Bethe mapping over $\mathbb{Q}\_p$ for $p\equiv 1 \(\rm{mod} \ 3)$. In fact, for $0 < |\theta-1|\_p < |q|\_p^2 < 1$, there existsa subsystem that isometrically conjugate to the full shift on three symbols.Meanwhile, for $0 < |q|\_p^2 \leq |\theta-1|\_p < |q|\_p < 1$, there exists asubsystem that isometrically conjugate to a subshift of finite type on $r$symbols where $r \geq 4$. However, these subshifts on $r$ symbols are alltopologically conjugate to the full shift on three symbols. The $p$-adic Gibbsmeasures of the same model for the cases $p=2,3$ and the correspondingPotts--Bethe mapping are also discussed.Furthermore, for $0 < |\theta-1|\_p <|q|\_p < 1,$ we remark that the Potts--Bethe mapping is not chaotic when $p=2,\p=3$ and $p\equiv 2 \ (\rm{mod} \ 3)$ and we could not conclude the vastness ofthe periodic $p$-adic Gibbs measures. In a forthcoming paper with the sametitle, we will treat the case $0 < |q|\_p \leq |\theta-1|\_p < 1$ for all $p$.

Periodic $p$-adic Gibbs measures of $q$-states Potts model on Cayley tree: The chaos implies the vastness of $p$-adic Gibbs measures