What is an example of a pair of Hopf algebras $(A,B)$ with a surjective Hopf algebra map $\phi:A \to B$ such that $\phi$ does not admit a $B$-bi-comodule splitting $s:B \to A$? To be clear, the right $B$-comodule structure on $A$ is given by
$$
(\textrm{id} \otimes \phi) \circ \Delta\_A: A \to A \otimes B,
$$
where $\Delta\_A$ is the coproduct of $A$, and the left coaction is defined similarily.



Covariant splittings of Hopf algebra projections
I'll give an example "occuring in nature." It's not the simplest possible, but you can get a simpler one by removing the generators of degrees 3, 5, and 7, which don't feature in the argument.


According to results of Borel from 1954, the mod-2 homology Hopf algebra $$H\_9 = H\_\* (\mathrm{Spin}(9);\mathbb F\_2)$$ is the exterior algebra on one generator each of degrees 3, 5, 6, 7, and 15. The standard inclusion $i$ of $\mathrm{Spin}(9)$ in $\mathrm{Spin}(10)$ preserves this exterior algebra structure, but $H\_{10} = H\_\*(\mathrm{Spin}(10);\mathbb F\_2)$ has a new generator $u\_{9}$ of degree 9 such that $H\_{10}$ is a free module of rank two over $i\_\* H\_9$ with basis $\{1,u\_9\}$,
following from the collapse of the Serre spectral sequence of the fiber bundle $\mathrm{Spin}(9) \to \mathrm{Spin}(10) \to S^9$. The new generator $u\_9$ doesn't anticommute with the old ones as one might expect: giving the other generators the obvious names, one has


$$u\_6 u\_9 + u\_9 u\_6 = u\_{15}$$


in $H\_{10}$.


Particularly, the injection of left $H\_9$-modules $H\_9 \to H\_{10}$ does not split. For degree reasons, $u\_9$ would have to go to $u\_3 u\_6$ or zero under the splitting, but the product $H\_9 \otimes H\_{10} \to H\_{10}$ sends $$u\_6 \otimes u\_9 \mapsto u\_{15} + u\_9 u\_6,$$ while the product $H\_9 \otimes H\_9 \to H\_9$ sends $$u\_6 \otimes 0 \mapsto 0 \neq u\_{15} + 0 u\_6$$ and also $$u\_6 \otimes u\_3 u\_6 \mapsto 0 \neq u\_{15} + (u\_3 u\_6) u\_6.$$


A $H^\*(\mathrm{Spin}(9);\mathbb F\_2)$-comodule splitting of the cohomological Hopf algebra map $$i^\*\colon H^\*(\mathrm{Spin}(10);\mathbb F\_2) \to H^\*(\mathrm{Spin}(9);\mathbb F\_2)$$ would lead on dualization to a forbidden module splitting of the sort we ruled out in the previous paragraph, so $i^\*$ is an example of the type you wanted.


