I was trying to unravel comments under the question When is it true that the ring of global regular functions on a projective variety is just the base ring?. It is apparently being claimed that if a proper morphism of finite presentation between schemes $X\rightarrow S$ is flat and has geometrically connected & reduced fibers then the natural map $$ \mathcal{O}_S(S)\rightarrow \mathcal{O}_X(X) $$ is an isomorphism. Is it true? By Grothendieck's coherence theorem the map is finite but I am not sure what to do next.

Maybe I should also mention that in my book empty space is connected (this probably does not change anything). Actually, flat morphism locally of finite presentation is open https://stacks.math.columbia.edu/tag/01UA so as long as we assume that $S$ is connected the fibers will be non-empty.