I think it would be great to display some custom message when the suggestions list is empty.
One example can be found on the video games web site g2a.com.
I attatch a picture of the fuctionality.

EDIT: if there is a way to do this in the current version I would be pleased if you tell me how to achieve this.
"No results" message is not part of react-autosuggest. It's totally up to you how to implement it.
Here is one example.
Hope this helps.
Most helpful comment
"No results" message is not part of react-autosuggest. It's totally up to you how to implement it.
Here is one example.
Hope this helps.