@Benjamin - Yes, but you'd have to develop that yourself using this API. There are a lot of examples online and we have implemented this custom instant search for a few of our customers.
For your reference, please check these links.
or you can try JS instant search, just use this API as the source.
I hope this helps.
I bring up a question from our devs.:
We tried to make the search request on a different domain as the community.
Is it possible to configure this for the Community page?
Or is there some kind of specific "public Lithium API Endpoint", against which there would be no problem to make AJAX requests?
CORS can be enabled for specific domains/urls by Lithium Professional Services or Support.
Note that CORS is only supported for REST API urls and custom endpoints.