* FTSEurofirst 300, STOXX 600 up around 0.6 pct
* BOE keeps rates on hold, markets trim gains
* Mining, energy, auto stocks in demand (Recasts, adds quote and detail, updates prices)
By Kit Rees and Atul Prakash
LONDON, July 14 (Reuters) - European shares slipped from a three-week high on Thursday after the Bank of England surprised investors by keeping interest rates on hold.
The pan-European STOXX Europe 600 and the FTSEurofirst 300 indexes were both up around 0.6 percent, having climbed earlier in the session to their highest since June 23 - the date when Britons voted in a referendum to leave the European Union.
European shares eased after the Bank of England caught investors off guard by keeping rates unchanged, with the UK's FTSE 100 index turning lower to trade just 0.1 percent higher.
Most economists taking part in a Reuters poll had expected the central bank to halve its Bank Rate to 0.25 percent in order to cushion the economy from the shock of the Brexit vote.
"Although the Monetary Policy Committee (MPC) surprised many by remaining on hold, we still expect an easing of policy in the near future," Bill Street, head of Investments EMEA at State Street Global Advisors, said in a note.
The market has been recovering from a sharp, post-referendum sell-off partly on expectations of further stimulus from central banks; the FTSEurofirst index is down just 1.9 percent since the vote.
The beaten-down banking sector continued its recent upward journey, with the European sector index rising around 1.7 percent, helped by a 3.2 percent rise in shares of Deutsche Bank and a 4.4 percent rise in UniCredit.
However, the banking index is still down around 28 percent, the worst performing sector this year, on concerns about lenders' underlying health and dwindling margins.
Automobile stocks, particularly sensitive to economic conditions, were the top gainers, with the sector index rising 1.8 percent on the back of a 2 to 5.7 percent rise in shares of Rheinmetall, Fiat, Daimler and BMW. (Editing by Mark Heinrich)