We study maximal cliques in the collinearity graphs of Desarguesian nets, give some structural results and some numerical information. In particular, we show for Desarguesian nets that the set consisting of a point $x$ together with all its neighbors on a line $L$ (with $x$ not on $L$) is contained in a unique maximal clique $C_{x,L}$ and determine the sizes and automorphism groups of such maximal cliques $C_{x,L}$ in all cases.