The priest, Mina Aboud Sharween, was attacked in the early afternoon while walking in the Masaeed area in El Arish.
The shooting in the coastal city was one of several attacks believed to be by Islamist insurgents that included firing at four military checkpoints in the region, the sources said.
Saturday’s attacks on checkpoints took place in al-Mahajer and al-Safaa in Rafah, as well as Sheikh Zuwaid and al-Kharouba.
The violence follows attacks in which five police offers were killed in El Arish on Friday.
Mr Morsi’s Muslim Brotherhood has fiercely criticised Coptic Pope Tawadros, spiritual leader of Egypt’s eight million Christians, for giving his blessing to the removal of Mr Morsi and attending the announcement by armed forces commander General Abdel Fattah al-Sisi suspending the constitution.
Officials say Adly Mansour, Egypt’s interim president, has held talks with the army chief and interior minister following overnight clashes across the country that killed at least 30 people and left dozens injured.
Running battles between those supporting the overthrow of Mr Morsi and those demanding his reinstatement were most intense in Cairo on Friday where tens of thousands took to the streets. The two sides briefly fought for control of the northern entrances to Tahrir Square.
The Muslim Brotherhood’s leader Mohammed Badie addressed a mass rally in the Egyptian capital on Friday and called for troops to defy their generals and restore Mr Morsi to power. He is believed to be under arrest.
Mr Badie said the national leaders who announced the new government on Wednesday night did not represent their followers.
He singled out the Grand Iman of Al-Azhar University and the Pope of the Coptic Christian church, a provocative move in light of the country’s rising sectarian tensions.
“Millions will remain in the squares until we carry our elected president, Mohamed Morsi, on our shoulders,” he told a cheering crowd. “We are his soldiers and we defend him with our lives.”
He said he would not be deterred “by threats or detentions, or the gallows”. He urged soldiers to obey their supreme commander – President Morsi – rather than the defence minister, Gen Abdulfattah al-Sisi.
Photo: Coptic Pope Tawadros II tweeted his support for the anti-Morsi protesters in Egypt earlier this week