According to the Flickr page, this building is "Trinity Place Garage". https://www.flickr.com/photos/28976636@N07/16123824309
It used to be a cinema, which closed in 1965 - see here: https://www.flickr.com/photos/28976636@N07/9526063626
The buildings are long gone, as you can see from the 2nd Flickr link, retirement flats now stand on the site.
This link http://www.eastbourneherald.co.uk/news/local/remarkable-town-stalwart-dies-aged-85-1-1406310 is an obituary of a Mr. Denys Lambert, who "...joined his father at Clark and Lambert, a garage in Trinity Place, Eastbourne, which merged with Caffyns in the early 1980s."