Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Add olive oil and let heat for 1 minute. Add mushrooms and onion and cook until onion is soft, about 3 minutes. Add garlic, dill, and oregano; cook for 1 minute. Hit Cancel.
Stir chicken broth and wild rice into the pot. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 25 minutes. Allow 10 minutes for pressure to build.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 2 minutes. Unlock and remove the lid. Stir in barley and peas, put the lid back on, and let sit for 10 minutes. Remove the lid and let sit until barley is tender and mixture has thickened, about 5 minutes longer. Season with salt and pepper.