Rooney, 30, played in midfield rather than as a striker and came under fire following the shock defeat by Iceland.
"He's our talisman, he's our greatest goalscorer, he's experienced and you don't want that to walk away at this stage," Lampard told BBC Sport.
"I would want him in the team and in the squad. He has certainly got another tournament in him."
Rooney has scored a record 53 goals in 115 appearances for his country.
The Manchester United man was made England captain in August 2014 but has come under scrutiny for a number of disappointing displays.
He found the net from the penalty spot after just four minutes against Iceland but underperformed for most of the game in Nice as England went out of Euro 2016 at the last-16 stage.
In the wake of the defeat, Germany legend Lothar Matthaus said it was time for Rooney to retire from international football.
"Sadly, when his team needed him, Rooney was not that leader," he said.
"You need someone who can provide the impetus and rhythm. I thought Rooney could take the role but, when it mattered, he wasn't able to do it."
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"I think once we reflect a bit more, Wayne will remain a big player in the squad and will help with the development of the younger players around him," insisted Lampard, who won 106 caps for England.
However, the former Chelsea and West Ham midfielder thinks he needs to work out his best position for both club and country.
Under Louis van Gaal, Rooney began to play in midfield as well as a striker but that may change now Jose Mourinho is manager of Manchester United.
"I don't believe 31 is an age where you need to change position because of being 31, said Lampard, now playing in the United States with New York City.
"I think Wayne still has the pace, the ability and the knowhow to play as a striker, but the choice will obviously be between him and his managers."
Rooney has already said he would continue to be available for international selection.
England's next game is a 2018 World Cup qualifier in Slovakia on 4 September.
England should not be afraid of appointing another foreign coach as a successor to Roy Hodgson, said Lampard, who played under Swede Sven-Goran Eriksson and Italian Fabio Capello during his time with the national side.
"Ideally, as a proud Englishman, you'd love an English manager," he said.
"But I think, in the modern world, you have to accept that, if the options aren't there, you have to look abroad.
"It wouldn't bother me if a foreign manager took an England team forward and was successful, not at all."
England are looking for a new manager as Englishman Hodgson quit immediately after the defeat by Iceland.
Under Eriksson, England reached the quarter-finals at three successive major tournaments, while Capello led England to the 2010 World Cup and a 4-1 defeat by Germany in the second round.
"I think the best man for the job would be the starting point for picking the next manager," added Lampard, who would prefer a youthful boss, adding: "Over the last 10-15 years, when I was playing, we weren't setting a new tone.
"I would look to a younger, forward-thinking manager who has a real strong idea about how he wants to play, plays good football, generates a good spirit in the team and go that way."
But he believes it will not be a straightforward task for the Football Association.
"I can't give you the name, which is the problem, but they will have to have a confidence in the decision they make," said Lampard.
"They'll have to give him free reign to try to mould these good young players into a team that can fight at the top level."
Wayne Rooney still has a future with England despite their Euro 2016 failure, says Frank Lampard.