The answer seems to be yes.  I think this was proved by Kuhlmann, Pank, and Roquette, ["Immediate and purely wild extensions of valued fields"][1], Manuscripta math. 55 (1986), 39-67.  A short proof is given in Efrat's book on valuation theory, p. 203.

 [1]:http://link.springer.com/article/10.1007%2FBF01168612