Are there noncomputable functions with a finite search space?
Depends on what exactly you have in mind.

If the problem is to find an element with decidable (!) properties among a  recursively enumerable set $A$ (of finitely represented elements), then yes: if $A$ is finite, the problem is trivially computable. Just check every element.

If $A$ is finite but unknown, it's more interesting but still: the problem is computable, since there is an algorithm that checks the every element in $A$, even though we can't point it out. See this for an example.

If the identifying property is undecidable, then the problem may be¹ uncomputable. For instance, the search space for the question "Does TM M halt on input x?" is finite -- just two possible answers -- but the problem is (in)famously undecidable.




It's possible that the property as written down is not needed to identify the correct element.