I made analysis please let me know if I am correct

Suppose I have the following graph

and let those arbitrary edges included in a cycle be ux and vx.

Now, if I remove vertex x, I can still reach from U to V, via the connected component shown.

Similar argument holds if I delete vertex U or V. and I am still able to cover remaining vertices.

Hence, the vertex connectivity of such a graph is not one which makes it nonseparable.