The Graber-Harris-Starr theorem says that any family of smooth rationally connected varieties over a complex curve has a section. A natural analogue of this statement in mixed characteristic would be that every rationally connected variety over the maximal unramified extension of a p-adic field has a rational point. I will discuss a geometric approach to this problem, as well as a proof of this statement for del Pezzo surfaces (for ${p>3}$). This is joint work with David Zureick-Brown.