0722 GMT December 19, 2018
Trump said he would never drop out of the race to be president and would never let his supporters down, BBC reported on Sunday.
He has been under pressure after a tape from 2005 of him bragging about groping and kissing women was broadcast.
The latest to withdraw their support were former Republican presidential candidate John McCain and former secretary of state Condoleezza Rice.
McCain said Trump's comments "make it impossible to continue to offer even conditional support for his candidacy", while Rice said: "Enough! Donald Trump should not be President. He should withdraw."
New Hampshire Senator Kelly Ayotte said in a statement: "I cannot and will not support a candidate for president who brags about degrading and assaulting women," she said
Ayotte – who faces a competitive race for re-election – said she would not vote for Clinton but instead would "write in" Mike Pence, Trump's vice-presidential running mate, on her ballot paper.
Several other Republicans also said they would vote for Pence.
Trump himself stressed that there was "zero chance I'll quit", adding that he was getting "unbelievable" support.
And in a tweet, the Republican candidate said "the media and establishment want me out of the race so badly".
Trump's wife Melania issued a statement on Saturday saying: "The words my husband used are unacceptable and offensive to me."
She said her husband had "the heart and mind of a leader".
Pence said he was "offended" by Trump's video, but grateful he had expressed remorse and apologized to the American people. "We pray for his family," he said in a statement.
Hillary Clinton, Trump's Democratic election rival, called his comments in the tape "horrific".
In the recorded comments, which date back to 2005 when Trump was appearing as a guest on a soap, he said "you can do anything" to women "when you're a star".
The candidate released a video statement apologizing for the comments.
Trump's 2005 comments, posted by the Washington Post, overshadowed the release of transcripts of Clinton's speeches to private events, by the whistle-blowing site Wikileaks.
The candidate had married his third wife Melania a few months before the recording. She said on Saturday: "I hope people will accept his apology, as I have, and focus on the important issues facing our nation and the world."