Berkeley is the name of various places, in Britain, Ireland and the United States. Some of these are spelled Berkley. Berk[e]ley is also the surname of various families that come from one or other of them. People in the UK with that name, including the famous philosopher Bishop (George) Berkeley (1685-1753), traditionally pronounce it as if the first vowel were an '-a-': 'BAr-kli' (IPA: /'bɑːk lɪ/), and this is true of the places in Britain and Ireland. In the United States, it is usual to pronounce it as it is spelled, with a first vowel like that in 'bird': 'BUr-kli' (IPA: /'bɜːk lɪ/). (This spelling is echoed in the British insult 'berk'.) This is the pronunciation of the first campus of the University of California, which was named after the great Irish philosopher.
A similar feature of pronunciation, with variation between British and American forms of English of the realisation of the spelling '-er-' as 'AHr', can be seen in clerk, the proper nouns Hervey and Derby, and the military rank of 'sergeant'.