The fighting was the latest in a series of sporadic battles throughout the day as tensions remained high between the two groups.
But the level of violence remained below that experienced on Sunday when running street battles killed eight and wounded 100 across Gaza.
Earlier in the day, Ismail Haniya, the Palestinian prime minister and Hamas leader, said: “Gaza today is better, and moving towards calm.
“We reiterate to our people to be responsible, not to spread the circle of disagreements and conflict, and not to transfer events to other parts of the nation.”
But just hours after he spoke, Fatah gunmen marched through Rafah after evening prayers to protest against the Hamas-led government, witnesses said.
When they approached Hamas gunmen, a gunbattle broke out and five people were wounded, one seriously.
The fighting ended after local Hamas and Fatah officials called on their supporters to back down.
But minutes later, a new battle erupted in a Fatah stronghold on the other side of the town. Two people were killed and nine suffered bullet and shrapnel wounds.
Fatah officials said Hamas gunmen opened fire on Fatah supporters as they drove near a roadblock. Hamas said its fighters came under fire from the car and fought back.
Violence erupted earlier at Gaza City’s main hospital when relatives of one of the men killed on Sunday arrived to collect his body.
Fatah gunmen accompanying them fired on Hamas fighters patrolling the hospital. Hospital officials said that nobody was hurt.
In the northern West Bank city of Nablus, Fatah fighters shot at the bodyguards of Nassir al-Shair, the deputy prime minister, as they rode in a government car. Two bodyguards and one Fatah fighter were injured in the incident.
In Jericho, a Fatah gunman trying to enforce the general strike shot a shop owner in the head, seriously wounding him, Fatah officials said.
The al-Aqsa Martyrs Brigades, an armed wing of Fatah, threatened for the first time early on Tuesday to kill Hamas leaders.
The threat marked an escalation in the power struggle between Fatah and the ruling Hamas party.
In a statement sent to Reuters, al-Aqsa Martyrs Brigades said it would kill Khalid Mishaal, the exiled Hamas political chief; Said Siyam, the Palestinian interior minister and Yusuf al-Zahar, a senior interior ministry official.
The statement said: “We in al-Aqsa announce, with all might and frankness, the ruling of the people in the homeland and the diaspora, to execute the head of the sedition, Khaled Meshaal, Saeed Seyam and Youssef al-Zahar, and we will execute this ruling so those filthy people can be made an example.”
Mishaal is based in Damascus but Siyam and al-Zahar are in the Gaza Strip.
Mushir al-Masri, a Hamas legislator, said al-Aqsa was “pouring oil on the fire” between the rival groups.
He said Hamas would “not show mercy” if any of its leaders were attacked by what he called “the leaders of the internal coup”.