His death came just three days after he published his final two judgments as a High Court judge.
One of these was the landmark and complex ruling on whether a number of patents are essential for the 3G telecoms standard, in a dispute between Nokia and InterDigital.
The other was a case over the broadcast of football matches in a pub using a feed from overseas.
Pumfrey had been a High Court judge, specializing in IP cases, since 1997, following a career as a barrister during which he advised the UK government on patent issues.
He was appointed to the Court of Appeal in November 2007, but delayed his move there so that he could hear the Nokia v InterDigital trial.
He was the second judge with IP experience in the Court of Appeal, alongside Lord Justice Jacob. Some observers believed that his appointment reflected the greater number and prominence of patent cases in the courts, and the need for specialist judges to hear them at all levels.
However, as an appeals judge Pumfrey would have heard cases in all areas, and there is no guarantee that his replacement will also be an IP specialist.
The Judicial Appointments Commission has not yet responded to enquiries regarding the details of Pumfreys replacement.
In a statement published by the Judicial Communications Office, the Master of the Rolls, Sir Anthony Clarke, said: "Besides being a life enhancing character, he was a great expert in the world of patents and intellectual property and was on the verge of contributing to the life of the Court of Appeal in many different areas. He was only 56 years of age and had been a member of the Court of Appeal for less than two months. He will be sadly missed."
Pumfrey took part in Managing IPs debate on the impact of the Gowers Review of intellectual property policy, in January 2007.
Obituaries have been published in the Daily Telegraph and The Herald newspapers.
His funeral will be held at Temple Church in London at 4.30 pm on January 14.