A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let $𝒮$ be a nontrivial finite regular linear space and $G\le \mathrm{Aut}\left(𝒮\right).$ Suppose that $G$ is extremely primitive on points and let rank$\left(G\right)$ be the rank of $G$ on points. We prove that rank$\left(G\right)\ge 4$ with few exceptions. Moreover, we show that $\mathrm{Soc}\left(G\right)$ is neither a sporadic group nor an alternating group, and $G=\mathrm{PSL}(2,q)$ with $q+1$ a Fermat prime if $\mathrm{Soc}\left(G\right)$ is a finite classical simple group.