It is really weird to see The Doctor, any Doctor, holding a gun, let alone using one, but Peter Davison's Doctor takes the cake. It just looks so wrong, especially compared to his rather bland and peaceful costume. It almost doesn't seem real to see Davison skulk around the warehouse, gun in hand, and then standing there, pumping six rounds into a Dalek mutant. At least Davison does his best to handle the gun like it's diseased when giving it back to Sergeant Calder.
This is, though, to the best of my knowledge, the most The Doctor has ever fired a good, old fashioned pistol. The First Doctor handles many a gun as part of a running gag in The Gunfighters (and reveals that he has a private collection of guns in the same story), and the Fourth Doctor brandishes an unfired pistol in The Seeds of Doom before finally pulling a trigger in The Talons of Weng-Chiang. I'll defend The Doctor's decision to shoot the giant rat in the sewers of London. It was an unnatural, vicious beast who was terrorizing the sewers, and The Doctor was the only one in the sewer. He needed to defend himself.
The Fifth Doctor in this episode, though, is surrounded by soldiers armed with machine guns. The Dalek mutant is the size of a kitten. What more damage could The Doctor possibly have done by adding in his contribution to the barrage of bullets? But this is a sign of the way Doctor Who was heading at this point in the show's history, especially under the stewardship of script editor Eric Saward.
Saward also wrote Resurrection of the Daleks, which is full of guns and death. His prior Earthshock is equally bloodthirsty, and in both, The Doctor, who has previously stayed out of such situations, is seen actively firing guns. In Earthshock, he fires a Cyber-gun into the Cyber Leader's chest, which is shocking enough, but he's using a laser gun. Laser guns are one thing. They're make believe. Guns that fire bullets are real. And seeing The Doctor use real world weapons helps bridge the gap between fantasy and reality that probably should rarely, if ever, be crossed.