The region's Asayesh police force said it was investigating the attack in the city of Dohuk.
The region's autonomous Kurdish authorities said the attacker was Syrian and influenced "by terrorist ideology affiliated to Daesh", using the Arabic acronym for IS.
Kurdish media said the attacker used a cleaver.
According to a police source, the attack took place in a market and targeted the Akitu springtime celebrations observed by members of the Assyrian community to mark the first day of their calendar year.
"A 65-year-old woman was hit on the head" and suffered a haemorrhage that did not require surgery, Dohuk medical authorities said, adding that her condition was "stable".
Another 25-year-old man suffered a minor scalp wound, the authorities added.
Dohuk governor Ali Tatar told a press conference that "our security forces arrested the suspect... the investigation is ongoing".
Iraq's Christian population plunged from some 1.5 million before the fall of Saddam Hussein in the early 2000s to about 400,000, the majority having fled successive bouts of violence in the country.
The rise in 2014 of the Islamic State, which had conquered the city of Mosul in northern Iraq and made it their "capital" notably accelerated their exodus.
But even when IS held Mosul and its surrounding areas until its defeat in 2017, neighbouring Kurdistan was relatively spared from jihadist attacks.
Jihadist cells are still active in Iraq, sporadically attacking the army and police, particularly in rural and remote areas.
A recent UN report said government-led counter-terrorism operations have resulted in the deaths of nearly half of IS's senior leaders in Iraq.
However, it warned the group could use the unstable situation in Syria following the overthrow of longtime ruler Bashar al-Assad, a member of the Alawite minority, to reinforce its position.
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