Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this family and the point $P$ we can associate the Kodaira-Spencer map

$$ Tan_S(P) \to H^1(X,T_X).$$

Here $X $ is the fibre of $f$ over $P$.

Of course, if $f$ is a product/trivial family (so $\mathcal X = X \times S$) then the Kodaira-Spencer map is zero.

On the other hand, I can't prove that the same holds if all closed fibres of $f$ are isomorphic (i.e. $f$ is isotrivial). Therefore, I suspect that there might be examples of non-trivial isotrivial families with non-zero Kodaira-Spencer map.

Are there examples of such families of varieties? (Note that such a family if it exists is not trivial, locally for the fppf topology.)