Corbyn, elected last year with the overwhelming support from grassroots Labour members, has faced criticism from some Labour lawmakers that he had not done enough to convince millions of voters in the party's heartlands to back remaining in the EU.
"I regret there have been resignations today from my shadow cabinet. But I am not going to betray the trust of those who voted for me, or the millions of supporters across the country who need Labour to represent them," Corbyn said in a statement.
"Those who want to change Labour's leadership will have to stand in a democratic election, in which I will be a candidate."
After a stinging election defeat for Labour last year, Corbyn won the leadership thanks to grassroots support but he has struggled to win the backing of Labour lawmakers, many of whom find themselves far closer to the centre ground than him.
Britain is not due to hold a parliamentary election until 2020, but after Prime Minister David Cameron's announcement that he would resign after losing the referendum, many expect that one could now be called earlier by his successor -- putting pressure on Labour to present itself as a credible alternative.
"I urge you, because you are a decent man, to do the decent thing and take the only action that can avert potential disaster by stepping aside," senior lawmaker Chris Bryant said in a letter to Corbyn, which he posted on Twitter.
"If you do so I believe future generations will praise your selflessness. If you refuse to step aside I fear you will go down in history as the man who broke the Labour Party."
Bryant, who said he was stepping down from his role in charge of Labour's parliamentary business, said Corbyn's "ambivalent attitude" to the EU campaign had betrayed the Labour party and wider movement.
Since the referendum, two Labour lawmakers have submitted a motion of no-confidence in Corbyn, calling for his leadership to be debated at a meeting of the party's lawmakers on Monday, followed by a secret ballot.
Early on Sunday the Labour leader sacked his foreign policy chief, Hilary Benn, after Benn said he had lost confidence in the ability of Corbyn, who is on the party's far left, to lead Labour.
Over the course of the day 11 others -- Bryant, Gloria de Piero, Heidi Alexander, Lucy Powell, Ian Murray, Kerry McCarthy, Vernon Coaker, Charles Falconer, Lilian Greenwood, Karl Turner and Seema Malhotra - said they were stepping down from Corbyn's top team of 30 lawmakers, known as the "shadow cabinet".
Malhotra, a member of Corbyn's finance policy team who had introduced him at a speech on Saturday, said Labour needed to strengthen its influence at a time of such political and economic upheaval.
"We need to recognise that we do not currently look like a government in waiting," she said in a letter to Corbyn, which she also published on Twitter.
Several of the others also said in statements that they doubted Corbyn's ability to lead the party to victory in the next election.