By Lang's theorem, a complete valued field which is the fraction field of a discrete valuation ring with an algebraically closed residue field is quasi-algebraically closed (or $C_1$).
How much is known about the converse? Is there a criterion/almost exhaustive list of complete valued fields, which is quasi-algebraically closed? For example, is $\mathbb{Q}_p$ quasi-algebraically closed?
Any reference dealing with such examples will be most welcome.