We know that those wizards who were afraid of death and chose to stay behind can leave an imprint of themselves upon the earth, to walk palely as ghost. But is it applicable to wizards only?
Is there any canon or other information about a ghost house-elf or a ghost goblin or ghost centaur?
It is always hard to prove a negative, but based on the evidence in the books and supplementary materials there is no mention of ghosts being anything other than (human) wizards.
When Harry is discussing death with Nearly-Headless Nick in Order of the Phoenix, Nick says,
"Wizards can leave an imprint of themselves upon the earth, to walk palely where their living selves once trod... But very few wizards choose that path."
Pottermore (now Wizarding World) also states that only wizards can becomes ghosts,
In the world of Harry Potter, a ghost is the transparent, three-dimensional imprint of a deceased witch or wizard, which continues to exist in the mortal world. Muggles cannot come back as ghosts, and the wisest witches and wizards choose not to. It is those with ‘unfinished business’, whether in the form of fear, guilt, regrets or overt attachment to the material world who refuse to move on to the next dimension.
However horses have been known to accompany their ghostly riders as well, as seen in Chamber of Secrets during the introduction of Headless Hunt,
Through the dungeon wall burst a dozen ghost horses, each ridden by a headless horseman. The assembly clapped wildly; Harry started to clap too, but stopped quickly at the sight of Nick’s face.
