The attacks came on Friday after a night of cross-border strikes in Gaza and Lebanon, added to heightened Israeli-Palestinian tensions following Israeli police raids in Jerusalem's Al-Aqsa mosque this week.
The tensions threatened to spiral into a wider conflict overnight as Israel responded to a barrage of rockets by hitting targets linked to the Islamist group Hamas in Gaza and southern Lebanon, but the fighting entered a lull on Friday.
However, the two attacks underlined how volatile the situation remains after successive nights of trouble that have drawn worldwide alarm and calls for calm.
In the latest attack, a car ploughed into a group on a street near a popular bike and walking path on a Tel Aviv promenade. The driver was shot dead by a nearby police officer when he tried to pull a gun, police said.
An Israeli security source identified the assailant as an Arab citizen of Israel from the town of Kafr Qassem.
The Magen David Adom ambulance service said the victims were all foreign tourists. Italy's Foreign Minister Antonio Tajani confirmed that an Italian had been killed and other Italians may have been among the wounded.
Earlier on Friday two Israeli sisters, aged 20 and 16 with joint British nationality, were killed and their mother wounded in a shooting attack on their car near the Jewish settlement of Hamra in the Jordan Valley.
"Our enemies are putting us to the test again," Prime Minister Benjamin Netanyahu said after visiting the site of the attack with Defence Minister Yoav Gallant.
As soldiers hunted for the gunman, Netanyahu ordered border police reserves and additional military forces to be mobilised to confront the wave of attacks.
The US State Department condemned the attacks, saying "the targeting of innocent civilians of any nationality is unconscionable".
No claim of responsibility was made for either of Friday's attacks, but Hamas, the Islamist group that controls the blockaded Gaza Strip, praised them and linked them to the tensions around Al-Aqsa mosque.
Friday prayers passed without major incident and apart from some stone-throwing, police said the situation had been quiet.
Twice this week Israeli police have raided the mosque, where hundreds of thousands of worshippers have been praying during the Muslim holy month of Ramadan, to dislodge groups they said had barricaded themselves with the aim of causing trouble.
Footage of officers beating worshippers who confronted them aroused concern, even among Israel's allies, and prompted condemnation across the Arab world.
The site in Jerusalem's Old City, holy to both Muslims and Jews, who know it as Temple Mount, has been a longstanding flashpoint, notably over the issue of Jewish visitors defying a ban on non-Muslim prayer in the mosque compound.
Clashes there in 2021 helped set off a 10-day war between Israel and Hamas. The exchange of cross-border fire awakened memories of that conflict, but as the lull in fighting extended on Friday, neither side seemed keen to prolong the fighting.
"Nobody wants an escalation right now," an Israeli army spokesman said. "Quiet will be answered with quiet, at this stage I think, at least in the coming hours."
One official with a Palestinian militant group told Reuters they were ready to keep the calm should Israel do the same, with the group having "made its point". A Qatari official said Qatar was helping international efforts to de-escalate the situation.
