Short answer
You should be mildly concerned, but it's not an emergency. Plain, cooked herring itself is safe for cats, but the creamy dill sauce is the problem — it likely contains ingredients like butter, cream, salt, and seasonings that can upset a cat's stomach. Watch your cat closely for signs of gastrointestinal distress, such as vomiting, diarrhea, or loss of appetite, over the next 24 hours. Pickled or heavily salted herring is unsafe, but a small amount of sauce-coated fish from one bite may just cause a mild tummy ache. Offer plenty of water and call your vet if any worrying symptoms appear.
Herring — Food safety facts
Safe to share · Cats · full breakdown on the linked food page
Cooked plain herring is fine. Pickled or salted herring is not.
The full verdict, portion guidance and serving notes are collected on the Herring food facts page.