The U.K. leader reiterated that she will not overturn the result of the 2016 referendum and "nor will I break up my country."
Speaking from inside Number 10 Downing Street, May repeated that the options on offer from the EU were unacceptable and would "make a mockery of the referendum we had two years ago."
May also called on the EU to provide more detail on its negotiating position.
"Yesterday, (European Council President) Donald Tusk said our proposals would undermine the single market. He didn't explain how in any detail or make any counter-proposal. So we are at an impasse."
In an apparent jab at the tone of other European leaders during negotiations, May added that healthy future relations were at risk.
"Throughout this process, I have treated the EU with nothing but respect. The U.K. expects the same. A good relationship at the end of this process depends on it," she said.
Sterling had fallen almost 1 percent in trading Friday prior to the speech, and was down to 1.3 percent at the end of May's statement.
After a dinner in Salzburg, the EU leaders said would continue to press for a Brexit deal next month but rejected May's proposal to seek a free trade area for goods with the EU.
EU leaders further cautioned May that talks could collapse if she didn't give leeway on the issues of Northern Ireland's border and trade.
After those warnings, May faced reporters Thursday evening and repeated that Britain was prepared to walk away from the EU without a deal. She further dismissed any suggestion that if the U.K. parliament was to reject her plan for leaving the EU, there could be a second referendum.
"There will be no second referendum," she said.
Under May's "Chequers plan," a U.K.-EU free trade area would be created with a "common rulebook" for industrial and agricultural goods. Britain's rule book on the goods would "harmonize" with that of the European Union.
It also calls for a "facilitated customs arrangement" to allow the smooth movement of goods between the U.K. and EU countries after Brexit.
Britain is due to leave the EU on March 29, 2019 and will then have 21 months to make the transition to full withdrawal at the end of 2020.