Your dog finds a stick and grips it harder when you approach. They stiffen or growl. Stop reaching and give space. Taking it by force risks a bite and may make the dog swallow a piece to keep it. If they are chewing or have swallowed splintered wood, contact your veterinarian with what you saw, choking, breathing difficulty, vomiting, or obvious pain need urgent care.
If the stick is not an immediate hazard and the dog is settled, avoid turning it into a struggle. You may be able to call the dog away and scatter treats at a safe distance, then retrieve the item only after the dog is securely separated. Do not try that if approaching would put you at risk. For a dog with repeated guarding or a prior bite, seek veterinary and qualified, reward-based behavior help.
Prevent the next contest
Walk around piles of sticks and use a secure lead in areas where pickups are likely. Practice a voluntary swap at home with low-value, safe toys, sometimes returning the item. Teach "leave it" before the dog takes an object, beginning with easy examples. Do not rehearse either cue with a guarded stick outdoors until a professional helps assess the risk.
The goal is not to show the dog that you can always take what they find. It is a safer walk where they are less likely to pick up a hazard and do not need to defend it. For more on preventing conflicts over found objects, see the resource-guarding guide, don't put your hand near a dog guarding a stick
Sources: Dogs Trust: Resource guarding in dogs. VCA Animal Hospitals: Ingestion of Foreign Bodies in Dogs.