LONDON, June 5 (Reuters) - British house prices unexpectedly fell in May, according to data from mortgage lender Halifax on Friday, which represented the latest sign of a cooling in the market as higher borrowing costs and uncertainty caused by the Iran war weigh on demand. House prices fell by a monthly 0.1% in May, the same as in April, representing the third consecutive month-on-month fall, Halifax said. A Reuters poll of economists had pointed to a rise of 0.1% on the month.