Set your Instant Pot to "Sauté" mode. Add olive oil to the pot and sauté the onions, carrots, celery, and garlic for 3-4 minutes until softened and fragrant.
Add the chicken breasts, diced potatoes, vegetable broth, thyme, salt, and pepper to the Instant Pot. Stir to combine.
Close the Instant Pot lid and set the valve to "Sealing." Select "Manual" or "Pressure Cook" mode and cook on high pressure for 10 minutes.
Once the cooking time is complete, quick-release the pressure and carefully open the lid. Remove the chicken breasts and shred them with two forks. Return the shredded chicken to the soup.
Add the dairy-free milk to the soup and stir well. If the soup is too thick, you can add more broth or milk to reach your desired consistency.
Stir everything together and taste for seasoning, adjusting salt or pepper as needed. Serve hot, and enjoy your creamy, comforting dairy-free chicken soup!