Taliban officials have said previously their forces had secured full control of Panjshir but fighting has been continuing for days, with each side saying it had inflicted large numbers of casualties.
Ahmad Massoud, leader of the NRFA, has pledged to continue resisting the offensive and has called for international support.
Panjshir, a rugged mountainous valley north of Kabul still littered with the wreckage of destroyed Soviet tanks, has proved very difficult to overcome in the past. Under Massoud's late father, Ahmad Shah Massoud, it resisted both the invading Soviet army and the previous Taliban government.
On Sunday, Massoud said hundreds of Taliban fighters had surrendered to NRFA forces, which included remnants of regular Afghan army and special forces units as well as local militia fighters. It was not possible to confirm that independently.
The Panjshir fighting has been the most prominent example of resistance to the Taliban, whose forces swept into Kabul on August 15 as the Western-backed government collapsed and President Ashraf Ghani fled the country.
But small individual protests for women's rights or in defence of the green, red and black tricolour flag of Afghanistan have also been held in different cities.
Massoud originally called for a negotiated settlement with the Taliban and several attempts at talks were held but they eventually broke down, with each side blaming the other for their failure.