  The quantum Yang-Baxter equation is a braiding condition on vector spaces
which is of high relevance in several fields of mathematics, such as knot
theory and quantum group theory. Their combinatorial counterpart are
set-theoretic solutions to the Yang--Baxter equation, whose investigation is
strongly driven by the study of algebraic objects called (skew) braces. In this
article, we focus on indecomposable involutive non-degenerate set-theoretic
solutions to the Yang-Baxter equation. More specifically, through a thorough
analysis of their associated braces, we give a full classification of those
which are of size $p^2$, for $p$ a prime.
