His mother, Nour El Houda Hassani, said: “Within him is a great secret. They killed him and buried the secret with him.”
Speaking in his impoverished Tunisian hometown of Oueslatia, she begged for his remains to be brought home.
“I want the truth about my son. Who was behind him?” she said.
Berlin Christmas market lorry attack
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Amri's brother Abdelkader wept as he said: "My brother is dead. Bring us his remains, even one of his fingers, and I will put it in my pocket.
“They killed him when he was still only a suspect. Why?"
The family want his remains to be sent to Oueslatia and asked authorities for details about his role in the attack.
Foreign Ministry spokesman Bouraoui Limam said he expected the body to be repatriated but that it would "take a while" because Italian, German and Tunisian investigators needed to carry out examinations first.
In the video, a man claiming to be Amri says: "My message to crusaders bombing Muslims everyday... Their blood will not go in vain. We are a nation behind them and will take revenge for them.
"I call on my Muslim brothers everywhere... Those in Europe, kill the crusader pigs, each person to their own ability.”
The Associated Press contributed to this report
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