We can work this out with a bit of lateral thinking
The short answer is simply NO, there aren’t mentioned characters in canon which can take the form of a magical creature.
The long answer is: I don’t think this is accidental.
In the series we see the Patronuses of only two people who are known Animagi: Minerva MacGonagall and James Potter. Surprisingly, both Patronuses match their Animagus forms.
If this tells us something about the correspondence between Patronus and Animagus then we can see what we know about the former to understand something about the mechanics of the latter. What we know about Patronuses of magical creatures? JKR has specifically stressed how Patronuses are quite always normal animals as opposed to magical creatures, except for very talented wizards
Extinct Patronuses are very rare but not unknown. Strangely, given
their long connection with wizarkind, owl Patronuses are unusual. Most
uncommon of all possibly Patronuses are magical creatures such as
dragons, thestrals and phoenixes.
(pottermore)
For example Albus Dumbledore’s Patronus is a Phoenix; also when discussing Remus Lupin’s Patronus she says:
Remus’s Patronus is never revealed in the Potter books […]. It is, in
fact, a wolf – an ordinary wolf, not a werewolf.
(pottermore)
Being an Animagus so rare, it’s not surprising how the few which managed to succeed are just regular animals… but if given these premises (Patronus form = Animagus form) we think of POTENTIAL ANIMAGI, we can SPECULATE that an exceptional wizard may be able to turn into a magical creature.
(However this may honestly lead to problematic questions: e.g. Dumbledore, whose Patronus is a Phoenix, would turn into a Phoenix himself? Would he be able to access to a Phoenix's magical immortality in this form? In my opinion this is unlikely to happen: it is possible (but it’s pure speculation not stated anywhere in canon) that the more magical the creature, the more the difficulty in becoming that animal [e.g. Niffler = limited magical abilities: higher probability of being a legit Animagus form; Phoenix = huge magical abilities: ability to take on this form practically equal to zero])
