On geometrically C₁ fields

A field k is called geometrically C₁ if every smooth projective separably rationally connected k-variety has a k-rational point. Given a henselian valued field of equal characteristic $0$ with divisible value group, we show that the property of being geometrically C₁ lifts from the residue field to the valued field. We also prove that algebraically maximal valued fields with divisible value group and finite residue field are geometrically C₁. In particular, any maximal totally ramified extension of a local field is geometrically C₁.