Civil war

You may be looking for the Gallifreyan conflict.

A civil war was a type of war between members of the same society.

The Dalek Civil War. (COMIC: Children of the Revolution)

Amongst humanity and others

On Earth, a number of civil wars were fought, including the English Civil War in the 17th century, (PROSE: The Roundheads) the American Civil War in the 19th century (TV: The War Games) and the Spanish Civil War in the 20th century. (PROSE: History 101) A time traveller claimed the Cybermen were caught in the Upgrade Wars in 2342. (PROSE: Time Traveller's Diary)

The Second Doctor admitted to Polly Wright that he had seen or heard of "all kinds of" possible futures from the "great" to "truly terrible", including Englands with a third, fourth, or fifth Civil War. (PROSE: The Roundheads)

Other civil wars included the Martian Civil War on New Mars, (AUDIO: The Prisoner of Peladon) the civil war which resulted in the fall of the Skonnon Empire, (TV: The Horns of Nimon) the Trion Civil War, (TV: Planet of Fire) the Rostax Civil War, (COMIC: Pay the Piper) and the City of the Saved Civil War. (PROSE: Of the City of the Saved, A Hundred Words from a Civil War, Tales of the Civil War)

Amongst the Daleks

The Daleks fought a number of civil wars: the Dalek Civil War between the Daleks loyal to the Dalek Prime and the humanised Daleks "infected" with the human factor, (TV: The Evil of the Daleks) the Imperial-Renegade Dalek Civil War between the Renegade Daleks loyal to the Dalek Prime and the Imperial Daleks loyal to Davros. (TV: Revelation of the Daleks, Remembrance of the Daleks) By one account, the Dalek Prime's Daleks reclaimed their original position as Imperial Daleks while those loyal to Davros were denounced as renegades. A new, albeit brief, civil war was waged as the Dalek Prime sought to eliminate the last of Davros' followers. (PROSE: War of the Daleks)

In what the Eighth Doctor described as a "meeting of the clans", a new race of Daleks created by Professor Martez on Red Rocket Rising were dismissed as mutants by the Daleks of Skaro, who saw to their extermination. (AUDIO: Blood of the Daleks) Similarly, the Death Squad Daleks identified and exterminated a New Dalek Army, created from Dalek DNA merged with human traces, on Earth in 2021. (TV: Revolution of the Daleks) During the Last Great Time War, a brief conflict broke out following the Dalek Time Strategist's resurrection of the Dalek Empire between those units loyal to the alternate Davros and the revived Dalek Prime. (AUDIO: Restoration of the Daleks)

Later in the conflict, Cardinal Ollistra attempted to cause civil war at every point in Skaro's history via the anima device only for the War Doctor to destroy the device, halting the scheme. (AUDIO: The Neverwhen)

In the Unbound Universe, the Doctor triggered a civil war among the Daleks when he reprogrammed one to think it was Davros, causing a schism between those Daleks who believed it and those remained loyal to the Black Dalek. The conflict came to a quick conclusion when the genuine Davros returned to Skaro, allied with the Quatch. (AUDIO: Masters of War)

Amongst the Time Lords

There were three civil wars on Gallifrey. Two were caused by Pandora attempting to seize power as Imperiatrix, (AUDIO: Lies, Imperiatrix, Fractures, Warfare) and a third was the result of the exile of Morbius. (PROSE: The Legacy of Gallifrey, Warmonger)

Behind the scenes

The Dalek Hierarchy feature of DWFC TLV 1 states that a civil war was fought between the New Dalek Paradigm and the "Children of Davros", which was glimpsed in The Eleventh Doctor Interactive Story. The Magician's Apprentice/The Witch's Familiar would depict a Dalek Empire favourable to Davros, who acknowledges them as his children, based on a rebuilt Skaro following, from the Twelfth Doctor's perspective, the Siege of Trenzalore, which, as established in Dalek: The Astounding Untold History of the Greatest Enemies of the Universe, saw the deaths of the Prime Minister as well as the original members of the New Dalek Paradigm during the Siege of Trenzalore. Indeed, no archetypes of the New Paradigm appear on the rebuilt Skaro.