  Let $A$ be an abelian variety over the function field $K$ of a compact
Riemann surface $B$. Fix a model $f \colon \mathcal{A} \to B$ of $A/K$ and an
effective horizontal divisor $\mathcal{D} \subset \mathcal{A}$. We study $(S,
\mathcal{D})$-integral sections $\sigma$ of $\mathcal{A}$ where $S \subset B$
is arbitrary. These sections $\sigma$ are algebraic and satisfy the geometric
condition $f(\sigma(B) \cap \mathcal{D})\subset S$. Developing the idea of
Parshin, we formulate a hyperbolic-homotopic height of such sections as a
substitute for intersection theory to establish new results concerning the
finiteness and the polynomial growth of large unions of $(S,
\mathcal{D})$-integral points where $S$ is only required to be finite in a thin
analytic open subset of $B$. Such results are out of reach of purely algebraic
methods and imply new evidence and interesting phenomena to the Geometric
Lang-Vojta conjecture.
