Penguin's Umbrella is an umbrella that the Penguin carried with him, but it was never a normal umbrella, and had built-in weaponry such a gun.
Arkham Origins Incident
Penguin used his umbrella to torture Alberto Falcone and there were umbrellas in his office and there was a case of them with his top hat. Many umbrellas were gold and some were black. In the game, you could hear some of Penguin's Henchmen talk about why their boss carried an umbrella, and thought about what it could do like maybe it was a gun, a sword, or it had a poison tip. One henchman thought that when the umbrella was opened, it could hypnotize the victim when spun (A Easter Egg or reference to Danny DeVito's, the Penguin from Batman Returns when he pretended to hypnotize Max Shreck, when it was really just a distraction to fire blanks).
Cold, Cold Heart Incident
Arkham Origins Blackgate Incident
Penguin had a bribed guard rescue him and also had bring him his umbrella from the prisoner belongings lockup. Penguin then killed the guard with his umbrella and then left without it.
Assault on Arkham Incident
When the Suicide Squad came to the Iceberg Lounge for help from Penguin in order to get into Arkham Asylum, Penguin spotted Harley Quinn, he took out his umbrella, (which had a gun built into it), pointed it at her, and explained to Deadshot and the rest of the Suicide Squad about how Harley and Joker had stolen a truck that was full of cigarettes and crashed it into the harbor "just for a laugh." Deadshot managed to persuade Penguin from shooting Harley and Penguin gave him the documents to get into Arkham. Penguin then told the Squad that they could stay the night in the rooms above, and then he didn't want to see any of them on his turf again.
Arkham Asylum Incident
Batman could see a collection of Penguin's Umbrellas in a display case in the South Corridor at the Arkham Mansion.
Arkham City Incident
Penguin used his umbrella to kill GCPD Officer Best, who was in his gang as an undercover spy, and also used it to unleash a Titan-fueled inmate on Batman. After Batman defeated Penguin, you could see an officer holding Penguin's Umbrella in the Iceberg Lounge.
Arkham Knight Incident
Penguin's Umbrella was seen in a display case for Penguin along with a top hat in the Evidence Room at the GCPD Lockup.
In addition, if the Batmobile gets destroyed on the Tim Burton's Batman-based race tracks, the hypnotizing spirals based on Penguin's umbrella appear briefly.