Combine flour, salt, and pepper in a shallow dish. Dredge chicken breasts through flour mixture.
Turn on a multi-functional pressure cooker (such as Instant Pot®), select Saute function, and allow pot to heat up. Add oil. Add chicken and cook until lightly browned, 3 about minutes. Turn chicken over and add mushrooms. Cook, stirring mushrooms occasionally, until other side of the chicken is lightly browned, about 3 more minutes. Pour Marsala wine around chicken. Close and lock the lid, and simmer until chicken is no longer pink in the center and the juices run clear, about 5 minutes.
Remove chicken from the pot and place on a serving platter. Add butter to the pot. Turn off heat and stir butter into sauce until fully blended. Pour sauce over chicken breasts and serve immediately.
