Pour red wine over short ribs in a large bowl. Marinate in the refrigerator, 8 hours to overnight.

Remove ribs from marinade, saving marinade for later use. Season short ribs with salt and pepper. Coat 1 side of each short rib with flour.

Select the Saute setting on an electric pressure cooker (such as Instant Pot); add olive oil and butter. Cook and stir until butter melts, about 2 minutes. Add 3 short ribs to hot butter-oil mixture. Cook until bottom is browned, about 8 minutes. Transfer browned short ribs to a plate. Repeat cooking with remaining short ribs, reserving butter-oil mixture in the pot.

Cook and stir onion, carrots, garlic, and rosemary in the hot butter-oil mixture, scraping browned bits off the sides of pot, until onion is softened and starting to brown, about 5 minutes. Add reserved wine marinade; stir well. Simmer until wine is reduced by 1/2, about 10 minutes.

Return ribs to the pot; add chicken stock. Ensure the total volume does not go above the maximum fill line and that the bulk of the ribs are submerged. Seal the pressure cooker. Turn the venting knob on top to Sealed, causing the floating valve to pop up.

Press the Manual button; set timer for 35 minutes. Tap the venting knob with a wooden spoon or spatula. Stand back; turn knob to point at Vent. Remove lid when pressure is released, about 5 minutes.

Transfer the ribs to a plate and cover with aluminum foil, reserving sauce in the pot.