S. G. Dani [Invent. Math. 64, 357–385 (1981; Zbl 0498.58013)] described the probability measures on the quotient \(G/\Gamma\) (where \(G\) is a connected semi-simple real Lie group and \(\Gamma\) is a lattice in \(G\)) that are invariant under the action of the unipotent radical \(U\) of a minimal parabolic subgroup of \(G\). Here the author proves a \(p\)-adic (and, more generally, \(S\)-arithmetic) analog of this result which, together with the work of M. Einsiedler and A. Ghosh [Proc. Lond. Math. Soc. (3) 100, No. 1, 249–268 (2010; Zbl 1189.37035)], constitutes a step towards the more general goal of finding \(p\)-adic analogs of the important theorems by M. Ratner concerning measure rigidity of unipotent dynamics [in: Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 157–182 (1995; Zbl 0923.22002)]. The results involve many technicalities, but the flavor of the results is contained in the following special case. Let \(G\) be a connected semi-simple algebraic group over \(K=\mathbb F_q(t)\) and \(\Gamma=G(\mathbb F_q[t])\), \(\bar{K}\) a completion of \(K\), and let \(U\) be the unipotent radical of a minimal parabolic subgroup of \(G(\bar{K})\). Then it is shown that any probability measure on \(G(\bar{K})/\Gamma\) that is invariant and ergodic for the action of \(U\) must be homogeneous in the sense that it is the translate of a measure arising from a parabolic subgroup. In particular, it follows that a \(U\)-orbit is uniformly distributed in its closure. The main results also apply to \(S\)-arithmetic groups of the form \(\prod_{v\in S}G(K_v)\) for a finite set \(S\) of places of \(K\).