I am not an expert on deformation theory, so I apologize in advance if the answer to this question turns out to be trivial.

I am currently reading the paper [1] and, a p. 737, I found the following remark that leaves me puzzled:

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. In fact, $\rho_{f, \, y}=0$ should mean that there is an analytic neighborhood $U$ of $y \in Y$ such that $f \colon \pi^{-1}(U) \to U$ is a trivial family. But in all the examples I know every fibre of $f$ is isomorphic at at most finitely many other fibres, hence (if I am not mistaken) the Kodaira-Spencer map should be injective at every point of $Y$. Probably I am missing something here, 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] Jabbusch, Kelly, Positivity of cotangent bundles, Mich. Math. J. 58, No. 3, 723-744 (2009). ZBL1186.14037.

[2] Catanese, Fabrizio, Kodaira fibrations and beyond: methods for moduli theory, Jpn. J. Math. (3) 12, No. 2, 91-174 (2017). ZBL1410.14010.