  We consider point sets in the affine plane $\mathbb{F}_q^2$ where each
Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets
are called integral point sets and were originally defined in $m$-dimensional
Euclidean spaces $\mathbb{E}^m$. We determine their maximal cardinality
$\mathcal{I}(\mathbb{F}_q,2)$. For arbitrary commutative rings $\mathcal{R}$
instead of $\mathbb{F}_q$ or for further restrictions as no three points on a
line or no four points on a circle we give partial results. Additionally we
study the geometric structure of the examples with maximum cardinality.
