The Sombor index, introduced by Ivan Gutman in 2020, has received intensive attention. The Sombor index of a graph $G$ is defined as $SO(G)=\sum _{uv\in E(G)}\sqrt{d_u^2+d_v^2}$, where $E(G)$ denotes the edge set in $G$ and $d_u$ denotes the degree of vertex $u$ in $G$. A graph with maximum degree at most 4 is called as a chemical graph. Réti et al. [T. Réti, T. Došlić, A. Ali, On the Sombor index of graphs, Contrib. Math. 3 (2021) 11-18] proposed an open problem about determining the maximum Sombor index among all connected $c$-cyclic graph for $6\le c\le n-2$. For $c=1,2,3,4$, the problem about finding the minimum (resp. maximum) Sombor index among all connected $c$-cyclic graph has already been solved. In this paper, we determine the minimum Sombor index among connected $c$-cyclic chemical graph for $c\ge 3,n\ge 5(c-1)$, which partially extends the results of Liu et al. [H. Liu, L. You, Y. Huang, Ordering chemical graphs by Sombor indices and its applications, MATCH Commun. Math. Comput. Chem. 87 (2022) 5-22] and Liu et al. [H. Liu, L. You, Y. Huang, Extremal Sombor indices of tetracyclic (chemical) graphs, MATCH Commun. Math. Comput. Chem. 88 (2022) 573-581].