LONDON, Feb 5 (Reuters) - British house prices fell last month for the first time since May, taking the annual rate of increase to its weakest since August, figures from mortgage lender Halifax showed on Friday. House prices dropped 0.3% on the month in January after holding steady in December, a weaker performance than the average 0.3% increase expected by economists in a Reuters poll. Compared with a year earlier house prices in January were 5.4% higher, slowing from 6.0% growth in December. (Reporting by David Milliken; editing by Michael Holden)