" "We'll kill him [Doc] tonight," Sam continued. "Headquarters has decided it's not worth it to bring him in for questioning. You two tail him for the rest of the day. Chances are he'll end up at the garage he uses for an office or at Twin Pines Mall. He's been spending a lot of time there recently, usually late at night." / "Does he carry any weapons?" [asked one of the Libyans] / "A handgun at most. An old .45-caliber revolver. It may not even work." "
"He [Marty] saw Doc Brown point the revolver at the [terrorists'] van and squeeze the trigger. No sound or flash of fire emerged, however, as bullets splattered all around Doc at his feet and into the side of the step-van. Finally, dropping the revolver, Doc began to sprint for the safety of the mall, fully five hundred yards away."
The weapon failed to fire when Doc also tried to use it against the Libyans in 1985, thus allowing the terrorists to shoot him dead.
Behind the scenes
In the novelization, Doc's handgun is described by Sam as being "an old .45-caliber revolver" (see Quote above).
On page 64 of the novelization (see second Quote above), a misprint resulted in the last part of the line No sound or flash of fire emerged, however, as bullets splattered all around Doc and into the side of the step-van reading 'bullets splattered all around Doc and into the side of the van' — implying that the terrorists' van was hit by the bullets instead, which (if one thinks about it) is absolutely impossible. This error has been corrected here.
There was a deleted scene where Doc offers Marty the handgun but he refuses and responds "No, thanks. I don't know how to use it anyway," which isn't true as he is a crackshot at Wild Gunman, He most likely refused as it could cause a paradox when Doc attempts to kill the Libyans in 1985, or because if he killed Buford Tannen in 1885 the timeline would change.
A deleted scene which shows Doc offering Marty his pistol.
Doc using the gun to start Marty in the DeLorean to go off to 1885.