Witnesses and medics said the first attack hit a car in the densely populated Rimal neighbourhood of Gaza City, setting the vehicle ablaze.
It was not immediately clear whether the five dead were passengers of the car or included passers-by.
Dozens of people rushed to extinguish the fire and rescue the victims.
Shortly after the attack on the car, the Israeli air force carried two separate air strikes on two houses in Deir al-Balah city and Nuseirat camp, in the central Gaza Strip, killing at least five people and wounding several others, medics said.
Later on Saturday a new Israeli air strike on a house in western Gaza City killed at least five Palestinians and wounded others, medics said, taking the day's death toll to at least 20.
The Israeli military said a gunman had crossed into Israeli-held territory in the Gaza Strip and exploited "the humanitarian road in the area through which humanitarian aid enters southern Gaza," calling it a "blatant violation" of the ceasefire agreement.
The military said it was striking targets in the enclave in response.
A Hamas official rejected the Israeli military's allegations and an "excuse to kill," saying the group was committed to the ceasefire agreement.
Israel and Hamas have repeatedly accused each other of violating the truce, concluded more than six weeks ago.
Hamas said in a statement earlier on Saturday that Israel's "escalating violations" put a responsibility on mediators and the US to confront it and preserve the ceasefire.
Israel also called on mediators to "insist that Hamas fulfil its side of the ceasefire" by returning the remaining three deceased hostages and completing its disarmament, Prime Minister Benjamin Netanyahu's office said.
The October 10 ceasefire in the two-year Gaza Strip war has eased the conflict, enabling hundreds of thousands of Palestinians to return to the enclave's ruins.
Israel has pulled troops back from city positions, and aid flows have increased.
But violence has not completely halted.
Hamas has been seeking to reassert itself and some are concerned about a de facto partition of the territory, where conditions are dire.
Palestinian health authorities say Israeli forces have killed 316 people in strikes across the Gaza Strip since the truce.
Israel says three of its soldiers have been killed since the ceasefire began and it has attacked scores of fighters.
