* Biggest fall for the pound in free-floating era
* Markets price in cut in Bank of England interest rates
* BoE pledges of extra funds, foreign currencies prop up FTSE
* Bank shares down (Recasts with FTSE recovering to gain on the week)
By Jamie McGeever and Patrick Graham
LONDON, June 24 (Reuters) - Sterling plunged to its lowest in three decades and the value of London's big banks sank by the most since the 2008 financial crisis as Britain's shock vote to leave the European Union triggered turmoil on global financial markets on Friday.
The damage to London's stock market eased as the day wore on, helped by expectations the weaker pound would help many UK companies and by the Bank of England's promise of 250 billion pounds of extra support.
But shares in Royal Bank of Scotland and Barclays fell by around 18 percent and, even with an afternoon recovery, sterling's fall was the biggest since the system of free-floating exchange rates was introduced in the early 1970s.
Major bank analysts predicted the currency would fall further in the months ahead, as financial investors price in the long- and short-term uncertainties unleashed by the Brexit vote and the scale of the damage to Britain's economic prospects.
"There is a grave danger of further weakness in the weeks ahead," said Societe Generale strategist Kit Juckes. "Indeed, the view of policymakers will be that a weaker pound is a vital economic shock absorber."
The moves dwarfed falls on "Black Wednesday" in 1992 when the pound was driven out of the pre-euro Exchange Rate Mechanism and London bankers who had worked through the night said it was the most volatile day's trading they had ever seen.
"The word 'unprecedented' is often used too much, and people often reach for the hyperbole. But this is truly unprecedented," said Steven Major, head of global rates strategy at HSBC in London.
HSBC cut its forecast for the pound to $1.20 and 92 pence per euro by the end of this year, and several other banks said they expected the value of the British currency to fall further.
Money markets moved fully to price in a cut in official interest rates by December, anticipating that the Bank of England will need to take steps to prop up an economy that has already slowed in the run in to Thursday's vote.
Traders said there had been strong buyers of sterling in Asia, however - possibly including foreign central bank reserve managers stocking up on the pound while it was at its cheapest in decades.
The cost of insuring against swings in the sterling/dollar exchange rate jumped to 53.375 percent, the highest since at least 1998, before easing back to less than half that.
Sterling stood at $1.3616 by late afternoon in London, up from as low as $1.3228, its weakest level since mid-1985.
"The pound fell a long way very quickly and the talk was that the speculative guys sold it on the first results last night and then bought it back," said Richard Benson, co-head of portfolio management at currency fund Millennium Global.
"At around $1.35 it looks to me to be over and done."
On stock markets, shares of Britain's biggest domestic banks took the biggest hit, but those of internationally-focused banks HSBC and Standard Chartered recovered to stand almost flat on the day.
Ten-year UK government bond yields dropped to 1.06 percent from around 1.38 percent late Thursday and Citi and Goldman Sachs both predicted a fall below 1 percent as investors took cover in the perceived security of government debt.
The biggest swings, however, were in the foreign exchange market, where trading went on through the night - albeit in light volumes for much of that time - and sterling tumbled to its lowest since before the signing of the Plaza Accord in 1985.
"I'm one of the people who was here the last time we were trading at $1.35. It's back to the future, we're back to where we were in 1985," said Nick Parsons, co-head of global currency strategy at NAB.
"We've had a 10 percent decline in six hours. That's simply extraordinary. And a vote to leave provides an existential crisis for Europe."
All the major international and British banks in London, including Citi Deutsche Bank, JPMorgan, Goldman Sachs and Barclays had traders either working through the night or on call.
On Citi's foreign exchange desk in London, dealers were only accepting voice orders and only desk heads had the authority to approve trades, according to a source at the bank.
Banks had warned clients about volatile trading conditions around the results which may lead to large gaps in prices. Barclays stopped accepting new "stop loss" orders on Thursday, an extremely rare move for one of the big six banks that dominate the world's biggest financial market. (Additional reporting by Vikram Subhedar, Richard Leong, Olivia Oran, Carmel Crimmins and Sinead Cruise; Editing by Andrew Heavens)