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Mr Farage said the party would campaign against "backsliding" on the UK's exit from the EU, saying he planned to see out his term in the European Parliament - describing his party as "the turkeys that voted for Christmas".
He said his party's "greatest potential" lay in attracting Labour voters, adding that he would not be backing any particular candidate to replace him.
"May the best man or woman win," he said.
Mr Farage said he would "bury the hatchet with anybody" including UKIP's sole MP Douglas Carswell, who tweeted an emoji picture of a smiley face as the leader's resignation was announced.
The two have repeatedly clashed and tensions rose when they backed rival Leave campaigns ahead of the referendum.
Mr Carswell told the BBC's Daily Politics Mr Farage had "played a role" in the EU referendum, but said his departure represented a "huge opportunity" for the party.
He described the chances of him running for leader as "somewhere between nil and zero" and said the winning candidate had to "steer UKIP away from the temptation of becoming an angry, nativist party."
Another of the party's high-profile figures, former deputy chairwoman Suzanne Evans, said she would "love to" become leader.
However, she is currently suspended from the party after an internal disciplinary meeting found she had publicly criticised a fellow candidate and held herself out as a party spokeswoman without authority.
Ms Evans told the BBC: "I've always said if Nigel ever stood down for whatever reason, I'd love to have the opportunity to lead UKIP but I'm in this rather bizarre situation that despite people wanting me to do that and wanting to do it myself, I'm actually technically suspended from the party at the moment until the end of September.
"So obviously if I'm not technically a party member, it's going to make it very difficult for me to stand."
'Want my life back'
Former Conservative minister Neil Hamilton, who leads UKIP's group in the Welsh Assembly, said he was surprised at the timing of Mr Farage's resignation and would not be standing to replace him.
"We would not have won this referendum but for the battle he has fought over the years," he added.
Mr Farage announced his decision to stand aside in a speech in London, saying: "I want my life back, and it begins right now."
He has been leader of UKIP for most of the past eight years, standing down briefly in 2009 and being re-elected the following year.
He said he would quit after failing to win his seat at last year's general election, but stayed on after the party rejected his resignation.