Flowers and candles have been placed outside the church in the southern French city of Nice where three people were stabbed to death.
President Emmanuel Macron said Thursday's stabbings were an "Islamist terrorist attack". Security is being stepped up throughout France.
The 21-year-old Tunisian suspect arrived in the city the night before the attack, his brother told the BBC.
Meanwhile, France's interior minister said more militant attacks were likely.
"We need to understand that there have been and there will be other events such as these terrible attacks," Gerald Darmanin said. "We're at war against an ideology, Islamist ideology."
Security has been increased at places of worship and schools across France following two similar attacks within two weeks. Earlier this month a teacher was beheaded in a Paris suburb after showing controversial cartoons of the Prophet Muhammad to some of his pupils.
Macron's subsequent defence of the right to publish the cartoons has stoked anger in several Muslim-majority countries.
Following the latest attack, police shot and wounded the suspected knifeman, who had only recently arrived in Europe. He is said to be in a critical condition in hospital.
Macron has ordered that the number of soldiers being deployed to protect public places - such as churches and schools - rises from 3000 to 7000.
Meanwhile, police investigating the attack have made a second arrest.
Two die inside church
The two women and a man were attacked inside the basilica in the morning before the first Mass of the day.
Two died inside the church. One of them, a 60-year-old woman who has not been named, was "virtually beheaded" close to the font, according to the French chief anti-terrorism prosecutor.
French media have named one victim as 55-year-old Vincent Loquès, a devout Catholic who had reportedly worked at the basilica for more than 10 years.
Loquès was opening the building when the attacker slit his throat, police say.
The third victim was named by the Brazilian foreign ministry as Simone Barreto Silva, a 44-year-old mother of three born in Salvador on Brazil's north-eastern coast. She had lived in France for 30 years.
She fled to a nearby cafe with multiple stab wounds but died shortly afterwards. "Tell my children that I love them," she told those who tried to help her, according to French media.
On Friday morning, priest Philippe Asso stood on the church steps with other mourners before walking in with a wreath to the victims. Others gathered outside the church to pay their respects.
Nice resident Frederic Lefèvre, 50, said he knew Loquès.
"This is a tragedy once again," he said. "We're a free country, we have demonstrated freedom to all countries of the world. Today, this freedom is closing in on us. Life needs to be lived for everyone."
Marc Mercier, 71, called the killings a "catastrophe".
"It's appalling. It's been years that we've been saying that fear should shift to the other side [attackers] but it is still the same."
Police sources named the man as Brahim Aouissaoui. Prosecutors said he had arrived by boat to the Italian island of Lampedusa as a migrant last month and after quarantining had been ordered to travel on.
He arrived in Nice by train and had no papers except for a Red Cross document from Italy, investigators said. A Tunisian official said he had not been listed as a suspected militant.
Speaking to the BBC at the family home near the Tunisian port city of Sfax, Brahim Aouissaoui's older brother, Yasine, said the pair had spoken on Wednesday night.
"He was sleeping the night before [the attack] on the stairs of some building in the area where he had been shot," he said. "He told me that in the morning he would try to find someone he knew to help him."
Yasine also described his brother as "a friendly person" who "never showed extremism". The suspect's mother, Gamra, said: "May God show the truth."
Witnesses said the attacker repeatedly shouted "Allahu Akbar" (God is greatest) before being shot by police.
A Koran, two telephones and a 30cm knife were found on him, French chief anti-terrorist prosecutor Jean-François Ricard said.
Two men have now been arrested by police investigating the attacks.
The first, a 47-year-old man believed to have been in contact with the suspect, was detained by police late on Thursday, French media reported.
The second, aged 35, was arrested yesterday. He had also had contact with Aouissaoui, police sources told AFP.
France has raised its national security alert to the highest level.
Anti-France protests growing
Meanwhile, thousands of Muslims in countries around the world have held protests to vent their fury at France over its perceived animosity towards Islam.
Anti-French rallies erupted after Friday prayers in Pakistan, Bangladesh, Lebanon and other countries.
Protesters called for a boycott of French products and condemned President Emmanuel Macron.
He has become a target in several Muslim-majority countries for his recent comments about Islam.
Macron has called Islam a religion "in crisis" and defended the right of a magazine to publish cartoons depicting the Prophet Muhammed.
The backlash intensified after a suspected Islamist extremist killed a French teacher who showed the cartoons in class. Macron said the teacher "was killed because Islamists want our future", but France would "not give up our cartoons".
Muslims saw Macron's remarks as an insult, as depictions of the Prophet Muhammad are widely regarded as taboo in Islam.
The protests continued amid heightened tension after the attack in Nice.