The Israeli military announced that one soldier had died from his wounds, the first in the most recent clashes.
Gaza officials said four Palestinians were killed in the strikes, three of them members of militant group Hamas.
They said a fourth Palestinian was shot dead during protests along the border. Hamas, who run Gaza, later said a ceasefire had been agreed with Israel.
"With Egyptian and United Nations efforts it has been agreed to return to the era of calm between (Israel) and Palestinian factions," Hamas spokesman Fawzi Barhoum told Reuters news agency.