Let $$M=S^n/ \Gamma $$ and h be a nontrivial element of finite order p in $$\pi _1(M)$$ , where the integer $$n, p\ge 2$$ , $$\Gamma $$ is a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form. In this paper, we prove that for every irreversible Finsler compact space form (M, F) with reversibility $$\lambda $$ and flag curvature K satisfying $$\begin{aligned} \frac{4p^2}{(p+1)^2} \left( \frac{\lambda }{\lambda +1} \right) ^2< K \le 1,\;\;\lambda < \frac{p+1}{p-1}, \end{aligned}$$ there exist at least $$n-1$$ non-contractible closed geodesics of class [h]. In addition, if the metric F is bumpy and $$\begin{aligned} \left( \frac{4p}{2p+1}\right) ^2 \left( \frac{\lambda }{\lambda +1}\right) ^2< K \le 1,\;\;\lambda <\frac{2p+1}{2p-1}, \end{aligned}$$ then there exist at least $$2\left[ \frac{n+1}{2}\right] $$ non-contractible closed geodesics of class [h], which is the optimal lower bound due to Katok’s example. For $$C^4$$ -generic Finsler metrics, there are infinitely many non-contractible closed geodesics of class [h] on (M, F) if $$\frac{\lambda ^2}{(\lambda +1)^2} < K \le 1$$ with n being odd, or $$\frac{\lambda ^2}{(\lambda +1)^2}\frac{4}{(n-1)^2} < K \le 1$$ with n being even.