Yes. In fact, this was a major story arc in the classic series. In a nutshell, there were two factions of Daleks: the “Imperial” Daleks, loyal to Davros, their creator, and the “Renegade” Daleks, who saw Davros as an inferior and unworthy leader. This plot arc was followed from the Fourth to the Seventh Doctors.
A detailed explanation is below. (Spoilers follow.)
In Genesis of the Daleks (Fourth Doctor, 1975), we witness the creation of the Daleks at the hand of Davros. Set on the planet Skaro, a race called the Kaleds are fighting another race called the Thals in a nuclear war. Davros was a Kaled scientist who built the Dalek travel machines as a way of destroying the Thals.
Davros attempts to control the Daleks, but of course they resist. He attempts to destroy them, but they realise what he’s doing and exterminate him first. This could be seen as the first act of a Dalek civil war.
In Destiny of the Daleks (Fourth Doctor, 1979), the Daleks are fighting a war with another race (the Movellans), who have unleashed a disease against the Daleks. They return to Skaro to revive Davros, and to use his scientific expertise to find a cure. This episode concludes with Davros being placed in cryogenic suspension, to be taken to Earth to stand trial for war crimes. (No major development of the civil war plot line, just setting the context for the next three episodes.)
The next three Daleks serials, which were the only serials between Destiny and the end of the classic series, centred around the idea of a Dalek civil war.
In Resurrection of the Daleks (Fifth Doctor, 1984), the Daleks break Davros out of a space prison to again assist in their war against the Movellans. The breakout attempt is lead by the Supreme Dalek, a role usually depicted in black (cf. Dalek Sec in the new series). As Davros works, he slowly builds a new, stronger army, and unrest builds in the Dalek camp. Some of the Daleks are loyal to Davros, while others are loyal to the Supreme Dalek. This episode concludes with Davros attempting to destroy the Daleks who are not loyal to him, by releasing the Movellan virus. However, he succumbs to it himself and is apparently killed.
(Trivia: in this episode, we see the Doctor handling weapons. He uses a pistol to kill a Dalek mutant. A reminder that the Doctor is not always as gun-averse as we like to think.)
Revelation of the Daleks (Sixth Doctor, 1985) was the next story in this arc. Taking place in a cryogenic facility, it turns out that Davros survived the Movellan virus (as a disembodied head in a tank), and is using the organic material at the facility to rebuild a Dalek army (not unlike Bad Wolf and The Parting of the Ways). The so-called Renegade Daleks intervene and battle the Imperial Daleks, eventually taking Davros back to Skaro to stand trial.
This episode also introduced differing colour schemes. The Imperial Daleks had a white-and-gold livery, while the Renegade Daleks were a more traditional grey-and-black. This Dalek spotters guide demonstrates the distinction well.
(Trivia: This was nearly the last Doctor Who serial ever made. The series instead had an eighteen month hiatus before returning, and left this episode on a cliff-hanger as to where the Doctor and his then-companion, Peri, would go next. Also the last serial to be made in 45-minute episodes before the reboot.)
The last episode in this arc, Remembrance of the Daleks, saw the two factions battling for a Timelord device called the Hand of Omega. Essentially a device that caused stars to explode, you can see why the Daleks would want it. Hostilities ensued, and the Imperials defeated the Renegades with the use of a Special Weapons Dalek, taking the Hand with them. But the Doctor destroyed the Imperials with the Hand of Omega, and convinced the Renegades to commit suicide, arguing that they no longer had a purpose.
This was the final Dalek serial in the classic series, and so the arc never went further than this. If we assume that the old series slides gracefully into the new, it’s reasonable to suppose that remaining hostilities were calmed in sake of the Time War, which posed a real threat to the continued existence of the Daleks. United against the Time Lords, old difference would be put aside.
We see hints of this rivalry in the new series, though. In Journey’s End, the Daleks are keeping Davros in the “basement” of the Crucible – clearly they still don’t trust him or want him getting too much power.
ETA: I realised I completely forgot to address the “not pure enough” comment in your question. This doesn’t really stem from the arc in the classic series, but it bears explaining. (Again, spoilers.)
In The Parting of the Ways, it’s revealed that the Emperor Dalek (who is pure Kaled/Dalek DNA) has been building an army of Daleks by harvesting human cells. These are not “pure” Daleks, but they nonetheless get the job done.
In Victory of the Daleks, a surviving handful of these impure Daleks recover a progenitor device which contains pure Dalek DNA. They use it to create the so-called New Paradigm Daleks, which then destroy the old Daleks because they are impure. The Daleks are focused on racial purity, so this isn’t so much a civil war as pruning the family tree.
Correction: as Amy points out below, it’s not Parting of the Ways that these Daleks come from, it’s Journey’s End. My mistake.
Indeed, the impure Daleks actually welcome this. They state, “We are ready” when the new Supreme announces their impending destruction. They serve the Dalek imperative (superiority and purity of Daleks), so they welcome their death since it purifies the Dalek line. (In a warped and twisted sense.)
Most of the above is cited from memory, so I may have made a few mistakes. Feel free to correct anything where necessary, but I think it’s accurate in broad strokes. If that’s not enough to read, then the Doctor Who Wiki has a short article on the Imperial-Renegade Dalek Civil War.
A lot of (non-canon) media continued story arcs after the television series was cancelled. Some of these may have the Dalek civil war arc, but I’m not aware of any that do. Exercise for the reader? :)
Finally, there was a long-running TV21 comic about the Daleks (I forget the exact dates). Not canon, but it proposed an alternative origins story and gave the Daleks more emotion and personality (imo). Since 2007, there’s been a CG comic called Second Empire, which tells the story of a Dalek civil war in the style of the old comics. The imagery and writing are excellent, it’s packed with laughs, and every time I read it I pick up new references. If you’re interested in the idea of a Dalek civil war, then I’d really recommend reading it. (It also uses the livery of the television arc described above.)
