Let D be a divison ring, let V be a left vector space of over D, possibly infinite dimensional, and let F be the prime field of D.

Is it true that every F-hyperplane of V contains a D-hyperplane of V?

I am mostly interested in the case when D has prime characteristic.

The answer is positive if D is finite, regardless of the dimension of V, although it is easier to see if V finite dimensional. Any help with with the case when D is infinite of prime characteristic would be appreciated.