Not strictly a Companion Chronicle in the first sense. There is not a Doctor Who companion telling the story or participating in it, but a notable character from one four-part story The Curse of Peladon. This is a Companion Chronicle in the other sense, a companion to the Doctor Who television chronicles.
Whilst we have not had a full television adventure without a companion to judge how the third Doctor would behave, here the author has a stab, an opportunity therefore to give us insight. Instead it rather detracts as the characterisation doesn't convince.
To a non-Who viewer, The Curse of Peladon is a well written story by more-than-compentent TV writer Brian Hayles, but less well executed because of the realisation of two creatures. One not so good creature (or its head) in a survival machine and one simply dreadful creature that is both awful in design and easily the worst voice in Doctor Who history. If this creature was a villain it would be deemed the worst, but it was not a villian. Unfortunately it appears in this story and the impersonation of the voice is very poor; but a very poor impersonation of a dreadful voice is two negatives and here the character is ironically more bearable than in the 1970s. The character is key to the plot so I can see why it is used, but frankly, some of the creatures in old Doctor Who annuals are better.
David Troughton deserves 5 stars, although a better use of him is in playing his father (Patrick Troughton) than as a one-off character. The format and plot themes are very similar to The Curse of Peladon, which whilst good from a familiarity point of view, something fresh would be just as welcome, as King Peladon, by his own rantings of never being told anything, is more two than three dimensional. The best thing about the Peladon stories were the glorious Ice Lord/Warriors, and I feel they were a little under-utilised here because the predominance of the narration is Troughtons tale telling, so 3 stars. Finally, a non-convincing interpretation of the third Doctor's persona - 2 stars. Therefore neither an awful nor brilliant product (which many Companion Chronicles are).
Overall, an interesting story but not a particularly welcome addition to the Doctor Who canon irrespective of ones' opinion of the Peladon stories. The Prisoner of Peladon is not a story that I can fully believe in, because of the Doctor's characterisation. Whilst it is not depressing or grey, it is one for completists of the Companion Chronicles or worshippers of the Peladon tales, otherwise there are more appealing or appropriate Big Finish audios for the occasional listener.