As far as I know, toric Fano manifolds are classified only up to dimension 4. In dimension one the projective line is the only example. In dimension two we have five examples: $\mathbb P ^2$, $\mathbb P^1 \times \mathbb P^1$ and $\mathbb P^2$ blown-up at 1,2 or 3 points in general position.
Toric Fano 3-folds were classified by K. Watanabe and M. Watanabe. Their result asserts that there are 18 types of such a manifold and that all of them have Picard number $\rho \leq 5$, the projective space $\mathbb P^3$ being the only one with $\rho =1$.
In four dimensions, V. Batyrev showed that there are 123 types of toric Fano 4-folds. In this case $\rho \leq 8$ and again the projective space is the only one with $\rho =1$
Classification in higher dimensions seems to be a rather diffcult problem, but the question whether $\mathbb P^n$ is the only toric Fano manifold with $\rho =1$ may be easier to answer. Is there any result in this direction?