Could Dumbledore's Animagus be a phoenix?

Are there cases of an individual whose animagus form is a magical creature?

We can figure this out with a little lateral thinking

The short answer is simple NO. There is no mention of characters in the canon who can take the form of magical creatures. The long answer is: I don't think this is coincidental.

In the series, we see the cartridges of only two people known to Animagi: Minerva MacGonagall and James Potter. Surprisingly both cartridges match their Animagus shapes .

If this says something about the correspondence between Patronus and Animagus, we can see what we know about the former to understand something about the mechanics of the latter. What do we know about patrons of magical creatures? JKR has specifically emphasized that unlike magical creatures, Patronuses are always normal animals, with the exception of very talented wizards

Extinct patron saints are very rare, but not unknown. Oddly enough, owl cartridges are unusual due to their long association with the sorceress. Most unusual of all possible cartridges are magical creatures such as dragons, thestrals, and phoenixes . (Pottermore)

For example, Albus Dumbledore's Patronus is a phoenix; even when she talks about Remus Lupin's Patronus, she says:

Remus' Patronus is never revealed in the Potter books [...]. It is indeed a wolf - an ordinary wolf, not a werewolf . (Pottermore)

Since it is such a rare animagus, it is not surprising that the few that have succeeded are just normal animals. However, if we take these requirements (Patronus shape = Animagus shape) into account, we can SPECULATE that an extraordinary wizard this can turn you into a magical creature .

(However, this can honestly lead to problematic questions: e.g. would Dumbledore, whose patronus is a phoenix, become a phoenix himself? Would he be able to access the magical immortality of a phoenix in this form? In my opinion, this is unlikely: It is possible (but it is pure speculation, not given anywhere in the canon) that the more magical the creature, the more difficult it will be to become that animal [e.g. Niffler = limited magical abilities: higher probability of being a legitimate form of Animagus to be; Phoenix = huge magic abilities: ability to take this form practically zero])