Nigel Farage has hinted that Douglas Carswell's future in Ukip is under threat over his calls for a "fresh face" at the top of the party.
Open warfare broke out between Ukip's senior politicians after Mr Carswell, the party's only MP, called on leader Mr Farage to quit.
Mr Farage said the current position could not continue and challenged Mr Carswell to "put up or shut up" - indicating that the party's ruling national executive could take a decision on his future in Ukip.
"I don't think our national executive will allow one individual to give an impression to the country that Ukip is divided when actually it is very united," Mr Farage said.
The leader claimed he enjoyed the "unanimous support" of the party's executive, MEPs and more than 90% of Ukip voters.
"We cannot have one individual ... every single time there is a Ukip conference it finishes with a story of disunity and it is all being caused by one person and, frankly, I think it is going to have to end," Mr Farage told BBC2's Daily Politics.
Asked if he thought Mr Carswell would quit the party, Mr Farage said: "I don't know what he wants to do. In the end it will not be me who makes these decisions, it will be Douglas who makes these decisions.
"We know that he agrees with us on the question of European Union membership. The difficulty is we think controlling immigration, having an Australian-style points system is the right way forward.
"He seems to feel it's a bit too awkward to talk about."
Mr Farage added: "I don't think our national executive will allow one individual to give an impression to the country that Ukip is divided when actually it is very united."
Mr Carswell's attack on the leader was motivated by the party's poor result in the Oldham West and Royton by-election, which saw Labour comfortably hold the seat with a majority of more than 10,000.
In an interview with BBC Essex, Clacton MP Mr Carswell said: "The Oldham by-election to me said very clearly that, I think, we need a fresh face."
Mr Carswell suggested that Mr Farage had taken the party as far as he could and Ukip needed to "change gear".
He said: "We all need to think very carefully as to whether or not we can build beyond the base we have now got without that change.
"We have gone from being a party with 2%-3% market share to a party with 13%, 14%, 15% market share, and that's a phenomenal achievement, an incredible achievement, and no-one can ever say that that hasn't been an achievement.
"But sometimes a start-up needs to change gear and to change its management if it is to go to the next level."
He said the party should avoid being seen as "unpleasant" and "socially illiberal".
Instead, Mr Carswell suggested Ukip would make a breakthrough in the polls if it was an "optimistic, sunshine, smiley, socially liberal, unapologetically free market party".
Mr Carswell, who joined Ukip in a high-profile defection from the Tories, ruled himself out of running for the leadership, saying he did not have the "patience", but insisted he remained loyal to the party.
"I am 100% Ukip and I'm very committed to Ukip, I'm not going anywhere," he said.
MEPs rallied around Mr Farage, with Ukip's deputy leader Paul Nuttall saying: " The Leave campaign needs a united Ukip. Nigel Farage has the support of the Ukip MEPs."
Roger Helmer said: "I wish that Ukip's Westminster parliamentary party would focus on the big issue of Brexit, not navel-gazing about internal party issues."
James Carver, the party's Commonwealth spokesman, tweeted: "Furious with Douglas Carswell's snipe at Nigel Farage, reported to coincide with the EU summit. Douglas, you're wrong. Ukip backs Nigel."
West Midlands MEP Bill Etheridge said Mr Carswell's comments were "unfortunate" and added: "I wish to make it absolutely clear that Nigel Farage has my 100% loyalty and support as leader of Ukip."
Ukip MEP Diane James told BBC Radio 4's World at One: "I'm absolutely astonished that Douglas has made the comments he has made today. His timing and his motivation I just can't even begin to fathom.
"Maybe he is part of a process and some form of organisation that wants to distract and divert from the fact that David Cameron achieved nothing in Brussels last night."
Asked if she was suggesting that former Conservative Mr Carswell might be acting as a "fifth column" inside Ukip for the Tories, Ms James said: "I do wonder at his motives.
"He has come from the Conservatives, there is a lot about him that seems to suggest that he is unhappy with the move he has made. Maybe he is reflecting on that.
"It is well known that he is still extremely friendly with existing Conservative MEPs."
Referring to the close associates of Julius Caesar who plotted to assassinate the Roman emperor, Ms James asked: "Is he the Brutus or the Cassius in the camp? That is the question that ought to be asked."
Ms James said: "You can't agitate in that way and keep undermining what the party is achieving and what Nigel is achieving without proposing an alternative.
"The ball is very, very, very firmly in Douglas Carswell's court - as in: Either put up or shut up or find an alternative."