Suppose that $G=\mathbb{S}^1$ acts freely on a finitistic space 𝑋 whose (mod 𝑝) cohomology ring is isomorphic to that of a lens space $L^{2m-1}(p;q_1,\ldots,q_m)$ or $\mathbb{S}^1\times\mathbb{C}P^{m-1}$. The mod 𝑝 index of the action is defined to be the largest integer 𝑛 such that $\alpha^n\neq 0$, where $\alpha\in H^2(X/G;\mathbb{Z}_p)$ is the nonzero characteristic class of the $\mathbb{S}^1$-bundle $\mathbb{S}^1\hookrightarrow X\to X/G$. We show that the mod 𝑝 index of a free action of 𝐺 on $\mathbb{S}^1\times\mathbb{C}P^{m-1}$ is $p-1$, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free 𝐺-action on $\mathbb{S}^1\times\mathbb{C}P^{m-1}$. It is note worthy that the mod 𝑝 index for free 𝐺-actions on the cohomology lens space is not defined.