Yes. The first was a pilot episode for a proposed Doctor Who radio series starring Peter Cushing that was recorded in the late 1960s; the pilot did not sell and the recording is now believed to be lost. In 1976 Argo Records produced Doctor Who and the Pescatons, an audio drama for LP starring Tom Baker and Elisabeth Sladen. In 1985 BBC Radio produced the serial Slipback starring Colin Baker and Nicola Bryant. Then, in the 1990s, BBC Radio produced two serials starring Jon Pertwee, Elisabeth Sladen and Nicholas Courtney (The Paradise of Death and The Ghosts of N-Space). In 2005, BBC Radio produced The Dalek Conquests, a retrospective of Dalek appearances up to the 2005 season, told in the form of an in-universe documentary.
Between 2008 and 2013, BBC Audio, later called AudioGO, produced a number of exclusive-to-audio stories, some presented as audio books (the first being the Tenth Doctor adventure Pest Control) and others more as regular dramas, including several multi-volume arcs starring Tom Baker and Richard Franklin. Throughout 2013, AudioGO also collaborated with Big Finish on an 11-part anniversary series titled Destiny of the Doctor which marked the first time Big Finish was allowed to use characters from the revival era.
In addition, there have been a number of BBC Radio dramas produced for Torchwood, as well as audio-exclusive audiobooks. The Sarah Jane Adventures has also been the subject of a number of audio-exclusive stories, e.g., "Judgement Day". SJA was in fact the first part of the franchise to have original-to-audio stories produced by AudioGO, predating Doctor Who and Torchwood.
Beyond Doctor Who proper, companies such as BBV have also produced "creator-licensed" audio dramas, which have not been authorized by the BBC and thus cannot feature or even mention the Doctor, but so long as the creators of a particular character have given permission, they are technically licensed. The Zygons are an example of characters who have been featured in such a manner.
AudioGO went out of business in 2013, but another company, Random House Audio, took over the license. As of February 2014 it has yet to be announced when or if it will be releasing further original audio content based upon Doctor Who or other aspects of the franchise.
See tardis:Category:Audio stories by company for a more extensive list of non-Big Finish productions.