Carswell's words are a direct attack on UKIP's leader Nigel Farage.
Speaking to the BBC's Nihal Arthanayake, who is making a documentary about him for BBC Essex, Carswell cited Farage's reaction to a recent by-election in Oldham as the reason that UKIP needs a fresh face.
Here is what Carswell said:
The Oldham by-election said to me very clearly that I think we need a fresh face. Let me put it another way, I don't want to wake up the morning after the European referendum and hear it was the postal votes.
BREAKING: @DouglasCarswell tells us exclusively it's time for @UKIP to have a fresh leader. #ukip pic.twitter.com/B9iwrLsJpp
- BBC Essex (@BBCEssex) December 18, 2015
UKIP were tipped to do well in Oldham but ended up coming second with 23% of the votes, while Labour won with 62%. Farage responded to the election defeat by claiming that the postal voting system was "bent."
Asked further by Arthanayake what he meant by a fresh face, Carswell was a little more guarded, saying that it was something that "we all need to think very carefully" about. Here's what he said:
It's not for me to decide who the new leader of the party is, but I think it's fair to say that we all need to think very carefully as to whether or not we can build beyond the base that we've got without that change.
WATCH: "We need a fresh face" @douglascarswell tells @therealNihal #UKIP needs a new leader, in BBC Essex exclusive. pic.twitter.com/eDWdavsZSi
- BBC Essex (@BBCEssex) December 18, 2015
BBC Essex asked UKIP deputy leader Paul Nuttall this morning whether Farage should step down. He said that it would be "insane" for Farage to do so and that Carswell was in a "minority of one." He then challenged Carswell to run for the leadership of the party - a contest he said Carswell would be humiliated in.
Here's Nuttall:
We will be speaking to Douglas today, finding out what the problems are, finding out what the issues are and let's see. If he's saying that he doesn't want to be the leader himself and he wants to put someone else forward, tell us who this person is. There's a mechanism within the UKIP constitution to force a leadership election, if Douglas wants to go down that road then so be it.
But I guarantee, that if he did, he would be humiliated because Nigel Farage would roll back in with probably a bigger proportion of the vote than what he got six years ago. So go ahead Douglas, put up or shut up and let's see what happens.
Farage also said that Carswell should "put up or shut up" on Sky News this morning, before saying that Carswell was "out on a limb and with almost no support."
The story becomes UKIP divided because one person frankly is out on a limb and with almost no support in the party and disagrees with the directional policy of the party.
Carswell defected from the Conservatives last year and became UKIP's first ever elected MP when he won the by-election he triggered by stepping down.
Carswell has become increasingly estranged from Farage in recent months, particularly over the approach UKIP should take as it campaigns for Britain to leave the EU in the upcoming referendum. Carswell is backing the Vote Leave campaign, while Farage is more aligned with Leave EU, which is bankrolled by multi-millionaire Ukip donor Arron Banks.