« Reply #4 on: Tuesday 10 May 05 03:27 BST (UK) »
Found it! 
It is listed in the 1911 Kelley's Directory for Sussex, with Henry Scott as the manager, with an address just given as Hampton Park (page 916).
In the 1965 Kelley's for Eastbourne it is shown as Hampton Park Hotel (Trust Houses Ltd) at 2 Brodrick Road H P. On modern maps Brodrick Road runs from near the railway station north then turns west towards Willingdon. It is one of those places that from a distance is "Eastbourne" but is on the outskirts of the modern town in what was the rural village fringe.
Rod
Jones, Ellis, Barker, Bates, Hackney, Cooper, Kirk, Eyre, Davies, Harris, Doney & Pearce.
Sussex, Cornwall, Lincolnshire, to name but a few!