Consider a fibration $F \rightarrow E \rightarrow B$ of "nice spaces" (e.g. compact connected smooth manifolds) with $B$ simply connected so we can ease spectral sequences computations. Suppose also that $H^*(E;\mathbb{Q})$ is a free $H^*(B;\mathbb{Q})$-module with the canonical structure induced by the map $E \rightarrow B$. (I will omit writing coefficients from now)
Using the Eilenberg-Moore spectral sequence, there is a spectral sequence
$$E_2 = Tor_{H^*(B)}(\mathbb{Q}, H^*(E)) \Rightarrow H^*(X) $$
By the condition on $H^*(E)$, the higher Tor terms vanish, then we have that the spectral sequence collapses at this page and $H^*(F) \cong \mathbb{Q} \otimes_{H^*(B)} H^*(E) $.
I want to show that $H^*(E) \cong H^*(B) \otimes H^*(F)$ as $H^*(B)$-modules. So we can use the Leray-Hirsch theorem for instance, but it would need that the map $i^*:H^*(E) \rightarrow H^*(F)$ induced by the inclusion of the fiber $i:F \rightarrow E$ admits an additive section.
my question is: Can we conclude from the isomorphism
$$H^*(F) \cong \mathbb{Q} \otimes_{H^*(B)} H^*(E) $$ that $i^*$ is surjective?