A Theorem of Faltings states that any proper subvariety of an abelian variety has finitely many rational points provided this subvariety does not contain a translate of any non-trivial proper abelian subvariety:

G. Faltings, The general case of S. Lang’s conjecture, pages 175–182 of Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math. 15, Academic Press, San Diego, CA, 1994.

Harris and Silverman use this to show that if C is curve of genus $\ge 2$ that is neither hyperelliptic nor bielliptic, then the set of rational points on the $C^{(2)}$ is finite. Here $C^(2)$ is the symmetric square of $C$.

J. Harris and J. H. Silverman, Bielliptic curves and symmetric products, Proceedings of the American Mathematical Society 112 (1991), no. 2, 347–356.

My understanding is that if $C$ has genus $3$ (say a plane quartic) then $C^{(2)}$ is a surface of general type.

I admit however that symmetric powers of curves are really special and probably not what the OP is hoping for.

A Theorem of Faltings states that any proper subvariety of an abelian variety has finitely many rational points provided this subvariety does not contain a translate of any non-trivial proper abelian subvariety:

G. Faltings, The general case of S. Lang’s conjecture, pages 175–182 of Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math. 15, Academic Press, San Diego, CA, 1994.

Harris and Silverman use this to show that if C is curve of genus $\ge 2$ that is neither hyperelliptic nor bielliptic, then the set of rational points on $C^{(2)}$ is finite. Here $C^{(2)}$ is the symmetric square of $C$.

J. Harris and J. H. Silverman, Bielliptic curves and symmetric products, Proceedings of the American Mathematical Society 112 (1991), no. 2, 347–356.

My understanding is that if $C$ has genus $3$ (say a plane quartic) then $C^{(2)}$ is a surface of general type.

I admit however that symmetric powers of curves are really special and probably not what the OP is hoping for.