A fixed point theorem for the action of linear higher rank algebraic groups over local fields on symmetric spaces of infinite dimension and finite rank

Let F be a non-archimedean local field of characteristic zero whose residue field has at least three elements. Let G be an almost simple linear algebraic group over F, with rank_F(G) >= 2. Let X be a simply connected symmetric space of infinite dimension and finite rank, with non-positive curvature operator. We prove that every continuous action by isometries of G on X has a fixed point. If the group G contains SL_3(F), the result holds without any assumption on the non-archimedean local field F. The result extends to cocompact lattices in G if the cardinality of the residue field of F is large enough, with a bound that depends on rank_F(G).