Governor Attaullah Khogyani said 13 other people, mostly women, were injured at the stadium in eastern Nangarhar province, where they were trying to get visas to enter neighbouring Pakistan.
He said most of those who died were elderly people from across Afghanistan.
In a separate incident, at least 34 Afghan police were killed in an ambush by Taliban militants in northern Afghanistan, according to a local hospital official.
Rahim Bakhish Danish, director of the main hospital in Takhar province, confirmed receiving 34 bodies and said another eight security force members were wounded.
An Afghan security official said the forces were in a convoy that was ambushed.
The official added that several police Humvees were set on fire.
Jawad Hijri, a spokesman for the provincial government, said the deputy police chief was among those killed.
There was no immediate comment or claim of responsibility from the Taliban, who control the area where the attack occurred.
The Pakistani Consulate in Nangarhar was closed for almost eight months due to the coronavirus pandemic. Anticipating a large crowd, officials decided to use the stadium and assigned 320 staff to help manage the process, Governor Khogyani said.
The Pakistani Embassy in Kabul said it has issued more than 19,000 visas in the past week alone after Islamabad approved a friendlier visa policy and reopened the border in September following months of closure.
Millions of Afghans have fled to Pakistan to escape war and economic hardship, while thousands travel back and forth for work and business, or to receive healthcare.
