Suppose we are over the complex numbers and that $f:S\to C$ is a smooth morphism from a smooth surface $S$ to a smooth curve $C$ of genus at least $2$, with non-constant classifying map $C\to M_g$, where $g$ is the genus of the generic fiber. (These exist, and are sometimes called Kodaira surfaces.) There is a tautological exact sequence $0\to f^*\Omega^1_C\to\Omega^1_S\to\Omega^1_{S/C}\to 0$; this is not split, since the classifying map is non-constant, and so has non-zero derivative, while $f^*\Omega^1_C$ is a sub-bundle of $\Omega^1_S$.

Now suppose that $\Omega^1_S = O(A)\oplus O(B)$. The de Franchis-Bogomolov lemma gives that each subsheaf $O(A), O(B)$ of $\Omega^1_S$ has Kodaira dimension at most $1$. On the other hand, $f^*\Omega^1_C$ embeds into each of $O(A)$ and $O(B)$ (else the tautological sequence splits). So we have $A\sim f^*K_C+D$ and $B\sim f^*K_C+E$ with $D,E\ge 0$. Since $K_C$ is ample on $C$, each effective class $D,E$ must be vertical (supported on fibers of $f$), by dF-B. But then $K_S\sim 2f^*K_C+D+E$ and so is also vertical, which contradicts the easy observation that $S$ is of general type.

[If these Kodaira surfaces seem too special, then use any semi-stable family of curves with base and fiber of genus at least $2$, and replace the bundles $\Omega^1_C$ and $\Omega^1_S$ by bundles of $1$-forms wioth appropriate log poles.]