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Published online by Cambridge University Press: 27 March 2020
Let 
$k$ be a finite field and 
$L$ be the function field of a curve 
$C/k$ of genus 
$g\geq 1$. In the first part of this note we show that the number of separable 
$S$-integral points on a constant elliptic curve 
$E/L$ is bounded solely in terms of 
$g$ and the size of 
$S$. In the second part we assume that 
$L$ is the function field of a hyperelliptic curve 
$C_{A}:s^{2}=A(t)$, where 
$A(t)$ is a square-free 
$k$-polynomial of odd degree. If 
$\infty$ is the place of 
$L$ associated to the point at infinity of 
$C_{A}$, then we prove that the set of separable 
$\{\infty \}$-points can be bounded solely in terms of 
$g$ and does not depend on the Mordell–Weil group 
$E(L)$. This is done by bounding the number of separable integral points over 
$k(t)$ on elliptic curves of the form 
$E_{A}:A(t)y^{2}=f(x)$, where 
$f(x)$ is a polynomial over 
$k$. Additionally, we show that, under an extra condition on 
$A(t)$, the existence of a separable integral point of ‘small’ height on the elliptic curve 
$E_{A}/k(t)$ determines the isomorphism class of the elliptic curve 
$y^{2}=f(x)$.