* Carney to stay on to end of January 2020
* Hammond says time is running out for a deal
* Sterling erases gains as optimism fades
* Boris and Brexiteers criticise May's plans (Recasts with new details throughout)
LONDON, Sept 11 (Reuters) - Mark Carney will stay on as Bank of England governor until the end of January 2020 to help navigate the turbulence of Brexit, Britain's finance minister said on Tuesday, as he warned time was running out to secure a Brexit agreement.
The United Kingdom is due to leave the European Union on March 29, but no agreement has been reached on the terms of its exit. Rivals to Prime Minister Theresa May are circling and some lawmakers are pushing for a rerun of the 2016 referendum.
Hammond said the government was devoting all its efforts to securing a deal on a new relationship with the European Union. Carney's continued presence at the central bank will help Britain negotiate the aftermath, he said.
"I have been discussing with the governor his ability to be able to serve a little longer in post in order to ensure continuity through what could be quite a turbulent period for our economy in the early summer of 2019," Hammond said.
Carney, a former Goldman Sachs banker, has run the Bank of England since 2013. He gained respect from investors for his handling of the effects of the Brexit vote.
Sterling, which tumbled after Britain voted to leave, has been moving sharply on Brexit news in recent weeks. The British currency reached its highest since early August $1.3087 early on Tuesday but gave up its gains to stand 0.2 percent, down at $1.3002.
As May tries to clinch a deal with the EU, she is facing rebels in her own party who have warned they will vote down any deal she makes unless she seeks a sharper break with the EU.
Michel Barnier, the EU's chief negotiator, said on Monday that a Brexit deal was possible "within six or eight weeks" if negotiators were realistic in their demands.
Hammond said Britain was working for a deal but that time was running out.
"We are working against the clock, we understand that," Hammond told parliament. "We will be working flat out over the coming weeks and months to achieve (a deal)."
Her spokesman said May saw an informal EU summit in Salzburg next week as a "staging post" in Brexit negotiations that would allow EU leaders to discuss her proposals for the first time.
Under May's proposals, Britain will seek a free-trade area for goods with the EU and accept a "common rulebook" for goods. The rebels say that will keep the UK under the EU's sway for years.
At a presentation by economists who favour Britain's leaving without a deal, Conservative lawmakers who support Brexit warned that May would have to ensure Britain was independent of the EU after Brexit.
Any joint EU-UK political statement, Steve Baker said, would have to "point towards an advanced free-trade agreement which leaves us as an independent country."
Boris Johnson, May's former foreign minister and a leading Brexit-supporting rebel, warned Britain should not accept single market legislation. . (Reporting by David Milliken, Andy Bruce, Alistair Smout and Elizabeth Piper, writing by Michael Holden Editing by Gareth Jones)