This question arose by reading the paper **[1]**, in particular, the remark at p. 737:

> As an example, consider a non-isotrivial smooth projective morphism $f \colon X \to Y$ from a smooth projective surface to a smooth
> projective curve. Suppose that there exists a $y \in Y$ such that the
> Kodaira-Spencer map $$\rho_{f, \, y} \colon T_{Y, \, y} \to H^1(X_y, \,
T_{X_y})$$ is zero. Then $\Omega_X|_{X_y}=\mathcal{O}_{X_y} \oplus
\Omega_{X_y}$, so $\Omega_X|_{X_y}$ is not ample.

Smooth, non-isotrivial fibrations from a surface to a curve are called *Kodaira fibrations* (see **[2]** for a good introduction to the subject) and the usual way to construct them is taking suitable ramified covers of a product of two curves. 

Now, I do not see how to perform the construction in such a way that the above condition on the Kodaira-Spencer map is satisfied. So let me ask the

> **Question.** What are examples of non-isotrivial smooth projective morphisms $f \colon X \to Y$ from a smooth projective surface to a
> smooth projective curve such that the Kodaira-Spencer map vanishes at
> some $y \in Y$?

**References.**

**[1]** <cite authors="Jabbusch, Kelly">_Jabbusch, Kelly_, [**Positivity of cotangent bundles**](http://dx.doi.org/10.1307/mmj/1260475697), Mich. Math. J. 58, No. 3, 723-744 (2009). [ZBL1186.14037](https://zbmath.org/?q=an:1186.14037).</cite> 

**[2]** <cite authors="Catanese, Fabrizio">_Catanese, Fabrizio_, [**Kodaira fibrations and beyond: methods for moduli theory**](http://dx.doi.org/10.1007/s11537-017-1569-x), Jpn. J. Math. (3) 12, No. 2, 91-174 (2017). [ZBL1410.14010](https://zbmath.org/?q=an:1410.14010).</cite>