Mayhew and Royle (2008) showed that there are 564 excluded minors for the class of GF(5)-representable matroids having at most 9 elements. We enumerate the excluded minors for GF(5)-representable matroids having 10 elements: there are precisely 2128 such excluded minors. In the process we find, for each $i \in \{2,3,4\}$, the excluded minors for the class of $\mathbb{H}_i$-representable matroids having at most 10 elements, and the excluded minors for the class of $\mathbb{H}_5$-representable matroids having at most 13 elements.