Support for the online campaign has soared in recent days after Theresa May agreed to pay up to £39 billion to leave the bloc.
Brussels politicians have caused further outrage among Leavers by saying Brexit trade talks cannot formally begin until March.
A petition, demanding Mrs May ditches the Article 50 process, sailed past the 100,000 mark on Saturday, meaning it must now be considered for debate by MPs.
By Monday morning it stood at just under 118,000 signatures, with the number shooting up every few seconds.
It was set up by Leave voter Jack Taylor, who wrote: “The Government should walk away from the Article 50 negotiations and leave the EU immediately with no deal.
“The EU looks set to offer us a punishment deal out of spite.
“Why wait another 18 months when we could leave right away and fully take back control of our country, lawmaking powers and borders?”
And UKIP, which has strongly supported the campaign, said the sheer level of support showed it was now time for politicians to listen to the people.
Gerard Batten MEP, the UKIP Brexit spokesman said “Parliament will have to debate ditching Article 50 and leaving on our terms, not the EU’s. “This is what should have happened the week after the Referendum. The genuine Leave MPs must now make a powerful case for immediate withdrawal.”
Party leader Henry Bolton said it showed a loss of patience with the drawn-out process “outside the elite blue in Westminster”.
He said: “This is great news, and it shows that despite the efforts of the switch back Labour party and a handful of malcontent Tory back-benchers that ordinary people of this country are still determined to get the Brexit they voted for.”
In its official response to the petition, the Department for Exiting the EU ruled out caving in to those demands.
They wrote: “The country voted to leave the EU, and this Government respects that.
“A smooth and orderly exit is in the interest of both the UK and the EU.”
The development comes as Theresa May appeared to have secured a rare moment of Tory unity by successfully moving talks on to trade.
She will tell MPs today that the country will seek to sign trade deals with countries around the world despite potentially being bound by EU rules for around two years after Brexit.
The Prime Minister says, even though the UK is leaving the single market and customs union in March 2019 she wants “access to one another’s markets” to continue “as now” during an implementation period.
The EU’s guidelines say that during any transition period the UK would have to comply with the bloc’s trade policy – preventing it from striking its own deals with other countries.
But Mrs May says the UK wants to sign agreements which would come into force after the “strictly time-limited” period has ended.
Mrs May and her senior ministers have also begun the process of thrashing out the Government’s plans for a post-Brexit UK-EU trade deal as Brussels indicated she may not get the “special partnership” she wants.
A meeting of the Brexit “war cabinet” was taking place on Monday, with a full Cabinet on Tuesday, as ministers formally consider the relationship the Government wants with the EU.
The petition can be signed online here.