Is there an example of a non context-sensitive language? I know $CSL\subset UL$ can be demonstrated by reduction to the absurd, but I've been trying to find a language that is in Type 0 ($UL$) and not in Context-Sensitive Languages ($CSL$).

Is there any language with such restrictions that isn't a Context-Sensitive language?
Type 0 grammars denote exactly recursively enumerable languages, and CSL = NSPACE[$n$] (e.g., Wikipedia).  Thus, your favorite proof of the Space Hierarchy Theorem gives an explicit language in, say, NSPACE[$n^2$] \ NSPACE[$n$], thus in UL \ CSL.