I would suggest Sliema or St Julian's as they are not resorts so are not closed down in December. Another option is Valletta the capital city although it tends to be busy during the day and then very quiet at night.

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We were in Malta twice during New Year and it was a wonderful atmosphere, I suppose there is the same for Christmas. Valetta was decorated with lights and Christmas Carols were broadcasted on the streets in tye evening. In my opinion Valetta is a charming place, and there are several restaurants and bars open late in the evening. Hotel Castille is in your price range.