The non-Archimedean Green--Griffiths--Lang--Vojta conjecture for commutative algebraic groups with unipotent rank 1

Let $k$ be algebraically closed field of characteristic zero, let $G$ be a commutative algebraic group over $k$ such that the linear part of $G$ is isomorphic to ${\mathbb{G}}_a$, and let $X$ be a closed subvariety of $G$. We show that the Kawamata locus of $X$ is equal to a Lang-like exceptional locus of $X$, and furthermore, we identify a condition on $X$ that implies that these loci are proper subschemes of $X$. We also prove the strong form of the non-Archimedean Green–Griffiths–Lang–Vojta conjecture for closed subvarieties of commutative algebraic groups where the linear part is isomorphic to ${\mathbb{G}}_a\times {\mathbb{G}}_m^t$.