  Let $K$ be an algebraically closed, complete, non-Archimedean valued field of
characteristic zero. We prove the non-Archimedean Green--Griffiths--Lang
conjecture for projective surfaces of irregularity one. More precisely, we
prove that if $X/K$ is a groupless, projective surface that admits a dominant
morphism an elliptic curve, then $X$ is $K$-analytically Brody hyperbolic. The
main ingredient in our proof is a theorem concerning the algebraic degeneracy
of non-Archimedean entire curves in projective, pseudo-groupless varieties
admitting a dominant morphism to an elliptic curve.
