Is there a way to show the facet filter, even if the Search API module returns no result?
2 Answers
Not really, the facets (both search api and solr) are made up from the results of the search, therefore if there are no results then it doesn't know what to populate the facets with.
Depending on your expected output you may be best creating a custom/views block to display some relevant content then re-write the urls to conform with the search query, acting like a search facet.
If you're using search_api_db, then you could use this patch :
https://www.drupal.org/node/2219897
It will add an option in your search server settings tyo allow a facet block to display facets even if there's no results !