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A vehicle carrying unidentified armed men stormed the entrance of the building where the family lived, opening fire upon some of the residents, the source added.
One family member rushed to save his family, but a landmine near the house exploded as his car drove over it, killing him and his child on the spot.
Attackers in North Sinai usually plant landmines around the areas they target as a trap for security forces attempting to rescue victims.
The family had earlier been displaced from the border city of Rafah, amid security measures taken by the army to counter terrorist attacks in North Sinai across the border with Gaza.
Violence
The reasons for targeting the family remains unknown.
No group has claimed responsibility for the attack.
Separately, but also in Arish, four police conscripts were injured late Wednesday when an explosive device went off near a police truck in which they were travelling. They were near a police station when it happened.
Last week, three policemen were killed and 10 people, including civilians, injured, in a suicide attack outside a police club in Arish.
The Islamic State terrorist network has claimed responsibility for the attack.
Security troops in the largely desert Sinai peninsula have come under repeated attacks from hardline jihadists since the 2011 uprising that toppled long-time dictator Hosny Mubarak.
The extremists stepped up their attacks after the military deposed Islamist president Mohammed Morsi in 2013, killing scores of security personnel.
The violence has mostly occurred in northern Sinai, prompting authorities to set up a kilometre-deep buffer zone there along the Gaza border.
