  In this paper, we prove that for every irreversible Finsler $n$-dimensional
real projective space $(\mathbb{R}P^n,F)$ with reversibility $\lambda$ and flag
curvature $K$ satisfying
$\frac{16}{9}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$ with $\lambda<3$,
there exist at least $n-1$ non-contractible closed geodesics. In addition, if
the metric $F$ is bumpy with
$\frac{64}{25}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$ and
$\lambda<\frac{5}{3}$, then there exist at least $2[\frac{n+1}{2}]$
non-contractible closed geodesics, which is the optimal lower bound due to
Katok's example. The main ingredients of the proofs are the Fadell-Rabinowitz
index theory of non-contractible closed geodesics on $(\mathbb{R}P^n,F)$ and
the $S^1$-equivariant Poincar$\acute{e}$ series of the non-contractible
component of the free loop space on $\mathbb{R}P^n$.
