Turn on a multi-cooker (such as Instant Pot) and select the Saute function. Heat olive oil; add ground beef and salt. Cook, breaking it into clumps with a spatula, until thoroughly browned, about 6 minutes. Transfer beef to a bowl. Wash the pot and return it to the machine.

Pour 1 1/2 cups of water into the pot. Set the metal trivet inside.

Arrange lasagna pieces over the bottom of a 6-inch springform pan. Ladle 1/3 the tomato sauce on top; add 1/3 of the cooked beef and 1/3 of the fresh spinach. Dollop 1/2 the mozzarella cheese on top. Repeat layers using 1/2 of the goat cheese. Repeat once more, finishing with remaining mozzarella and goat cheese on top.

Tent the top of the lasagna loosely with aluminum foil. Lower it gently onto the trivet. Close and lock the lid. Select high pressure according to manufacturer's instructions; set the timer for 20 minutes. Allow 10 to 15 minutes for pressure to build. Release pressure using the natural-release method according to manufacturer's instructions, 10 to 40 minutes. Unlock and remove lid.

Set oven rack about 6 inches from the heat source and preheat the oven's broiler.