Would a two-seat light fixed-wing aircraft with a landing speed of 20 knots and a top speed of 180 knots be technically possible?
At least one airframe close to these specs already exists, its a highly modified Wilga by the name "Draco". It can get off the ground in 120 feet and has a stall speed of ~37mph and cruises at ~180mph. With its epic 4000 ft/min climb rate it can get out of tight spaces fast as well. The only thing it lacks compared to your specs is the retractable gear as it was designed as a big gear bush plane. 



(source)



The same guy (Mike Patey) also built a pretty wild Turbolance experimental that has some epic short field and climb performance as well and its a retract. 

His overall secret sauce is effectively putting a PT-6 on the front end of a small airframe to add a huge amount of power to a light airframe.