Let D be a divison ring of prime characteristic p and let V be a left vector space of dimension > 1, possibly infinite.

Let M be a maximal subgroup of the additive abelian group V, i.e., a hyperplane of the F_p-vector space V.

Does M necessarily contain a non-zero subsppace of the D-vector space V?

There are several equivalent ways to ask this question, of which the above is perhaps the most elementary.

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