Fraidoon Obaidi, a Kabul district police chief, said Lisa Akbari was attacked as she left the complex gym at 8 p.m. Sunday.
Obaidi said the mullah was injured but declined to explain how or why. He's been arrested, Obaidi said, but he didn't provide information about any charges the mullah faces. The chief did not provide the man's name but said he is being interrogated.
Obaidi told CNN that investigators have seized documents from the mullah's residence that show he allegedly has links to terrorist groups, but the chief did not specify which groups.
The U.S. Embassy confirmed Akbari's death Monday.
"We offer our sincerest condolences to her family and loved ones on their loss. We are providing all possible consular assistance. Out of respect for the privacy of those affected, we decline further comment," an embassy statement read.
The mullah lived in the woman's housing complex, Afghanistan's TOLO News agency reported.
About 34 miles away, six NATO service members were killed
in a Monday motorcycle bomb attack in Bagram, Afghanistan, military officials told CNN. All six were American, said U.S. Defense Secretary Ash Carter.
A suicide bomber on a motorbike carried out the attack on a joint patrol of Afghan and coalition forces at about 1:30 p.m. in the Bajawri area of Bagram district, said Waheed Sediqi, a spokesman for the governor of Parwan province.
The area is close to the U.S. base in the Bagram district of northern Parwan province.
"We're deeply saddened by this loss," Brig. Gen. Wilson Shoffner said in a statement. "Our heartfelt sympathies go out to the families and friends of those affected in this tragic incident, especially during this holiday season."
The names of those killed were not immediately released.