The latest deaths took the nationwide toll from the violence to 20.
There has been growing disquiet about the law, which gives people from persecuted minorities from three neighbouring countries an easier path to citizenship, but not if they are Muslim.
Police have so far said that the day's unrest in Uttar Pradesh claimed 11 lives in total, including the child, who was trampled.
Uttar Pradesh police spokesman Shirish Chandra confirmed to AFP that the other 10 people died after being shot.
"Ten people were killed on Friday. All of them were bullet fatalities. We are looking into other cases," Chandra said.
The boy had died after 2,500 people including children joined a rally in the holy city of Varanasi, district police chief Prabhakar Chaudhary told AFP.
"When the police tried to quell the protests, these persons ran for cover and a stampede-like situation emerged, in which this boy died," Chaudhary said.
He added that police "exercised complete restraint against the crowds that engaged in attacking them with stones".
The Times of India said the boy was playing in a lane with a friend when they were trampled by a crowd being chased by police.
Two deaths were in Uttar Pradesh's Kanpur district, a spokesman for the district police chief told AFP.
Five other people in the state's districts -- two from Meerut, two from Muzaffarnagar and one from Bignor -- died from gunshot wounds, medical and police officials told AFP Friday.
A sixth death was confirmed by police in Firozabad city on Friday. The locations of the other deaths have not been revealed by authorities.
Critics say the law discriminates against Muslims and is part of Prime Minister Narendra Modi's Hindu-nationalist agenda, but he has repeatedly denied the claims.
Based on AFP
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