* CPI slows to +2.6 pct in June vs +2.9 pct in May
* First fall in CPI since October
* Factory gate price growth eases too
* Pound falls as investors see less rate pressure on BoE (Adds market reaction, comment)
LONDON, July 18 (Reuters) - British inflation unexpectedly slowed last month for the first time since October, dousing expectations among investors that the Bank of England might soon raise interest rates for the first time in a decade.
Consumer prices rose by 2.6 percent compared with a year earlier, the Office for National Statistics said on Tuesday, down from a nearly four-year high of 2.9 percent in May.
Economists had expected the rate to remain unchanged.
Sterling fell after the data, down by half a cent against the U.S. dollar, and British government bond prices jumped as the figures suggested the BoE was under little pressure to raise rates when it next meets in early August, despite concerns among some of its policymakers about rising prices.
The fall in inflation was the sharpest between any two months since February 2015, largely reflecting a fall in global oil prices, and there were also signs of slowing price pressure in factories.
"This is going to kill the chances of a rate rise in the short term. We will learn more about the Bank of England's thinking in a couple of weeks, but we can expect the calls for a rate rise to reduce to a whimper," Lucy O'Carroll, chief economist at fund managers Aberdeen Asset Management, said.
However, many economists have said they expect inflation to pick up again soon, adding to the strain on households which are seeing salaries rise more slowly than prices.
"Although this has taken pressure off the Bank of England, I still think inflation will exceed 3 percent later this year," Alan Clarke, an economist at Scotiabank, said. "There's lots of pressure still in the tank."
Britain's inflation rate has risen sharply since last year's referendum decision to leave the European Union which pushed down the value of the pound, making imports more expensive.
BOE LIKELY TO REMAIN PATIENT
The BoE has been taken by surprise by the speed of the increase this year. Its most recent forecasts saw inflation peaking at 2.8 percent later in 2017 while most economists have said they expect inflation to reach at least 3 percent.
The BoE did accurately predict that inflation in June would be 2.6 percent.
The BoE has so far chosen not to respond by raising rates, saying the Brexit hit to the pound is likely to be temporary.
However, some officials at the central bank signalled recently that a rate hike might be on the way. Three of the BoE's eight rate setters voted to raise rates in June although one of the dissenters has since left
Governor Mark Carney has said higher borrowing costs would probably be needed if the economy overcomes its slowdown of earlier this year and wages - which are lagging behind inflation - grow more strongly.
The BoE is due to announce its next decision on interest rates on Aug. 3.
A Reuters poll of economists published on Tuesday showed the BoE was expected to keep rates on hold throughout 2017 and 2018, as it waits to see if wages catch up with price rises and how the economy copes with the approach of Brexit.
The ONS said inflation was pushed down by fuel prices after a fall in global oil prices which was accentuated by a partial recovery in the value of the pound last month.
A drop in the cost of computer games and equipment, which pushed up inflation in May, was also a factor.
Retail price inflation - tracked by British inflation-linked government bonds - slowed to 3.5 percent in June from 3.7 percent in May.
The ONS said excluding oil prices and other volatile components, core consumer price inflation slowed to 2.4 percent. Economists had expected it would remain at 2.6 percent.
Data on factory gate prices suggested that the most intense pressure on consumer prices is easing.
Output prices rose at their slowest rate since December of last year, up 3.3 percent, in line with the Reuters poll.
Prices paid by factories for materials and energy rose at their slowest rate since September, up 9.9 percent on the year. (Writing by William Schomberg; editing by Susan Thomas)