MacDill AFB in TX has allowed authorized personnel to "transport and secure privately owned handguns in privately owned vehicles". So far, Dover AFB doesn't agree.
http://www.tbo.com/list/military-news/on-duty-but-unarmed-macdill-airmen-cant-have-personal-weapons-on-the-base-20160218/"In December, Col. David Benson, commander of the 7th Bomb Wing at Dyess, decided to allow authorized personnel to transport and secure privately owned handguns in privately owned vehicles within the legal boundaries of Dyess Air Force Base provided they possess a current and valid Texas Concealed Handgun License or a reciprocating state concealed carry license.
Benson’s decision does not permit carrying concealed weapons on base. And individuals wishing to transport a handgun must also have a valid Department of Defense identification card or Common Access Card and hold status as active-duty personnel, reserve personnel, National Guard personnel, retired military personnel, dependents of active-duty personnel or Department of Defense civilians."