Compact logic without the downward Skolem-Löwenheim property

Question

Lindström's theorem suggests the existence of a compact logic without the downward Skolem-Löwenheim property. My main source for this kind of matter is Barwise and Feferman's Model-Theoretic Logics. Unfortunately, the book does not make it easy for me to answer this question. More seriously, many matters in this book (for instance, Ziegler's work on model theory of topological spaces) is forgotten nowadays. Thus my question is: what are compact logics without the downward Skolem-Löwenheim property in which people are still interested in?