  Let $K$ be a 1-dimensional function field over an algebraically closed field
of characteristic $0$, and let $A/K$ be an abelian surface. Under mild
assumptions, we prove a Lehmer-type lower bound for points in $A(\bar{K})$.
More precisely, we prove that there are constants $C_1,C_2>0$ such that the
normalized Bernoulli-part of the canonical height is bounded below by $$
\hat{h}_A^{\mathbb{B}}(P) \ge C_1\bigl[K(P):K\bigr]^{-2} $$ for all points
$P\in{A(\bar{K})}$ whose height satisfies $0<\hat{h}_A(P)\le{C_2}$.
