I attended a seminar that was hosted by Immigration/TSA/FBI etc and they made it clear that provided the PRIMARY PURPOSE of a visa was still being upheld (as per JulianS comment) then flight training may be carried out without any further documentation.
However, some of the FAA flight examiners in the area still do not accept the fact and will not conduct an FAA flight test without seeing a M1 visa - not that this situation involves an FAA examiner.