You will not be able to argue that a ghost does not exist, because of course why should she simply believe you instead of her mother? You don't have the authority to know better. Avoid the conflict of your truth versus her mother's truth. Instead try and get on her mother's side: What you can do is explain to her why her mother believed in ghosts. That is understandable for a child and will allow her to differentiate herself from the believes of her mother. But think about this carefully and don't devalue her mother, instead make believing something wrong a good thing. For example, you could say that believing in ghosts helped her mother because it gave her strength to make a good life for her child. Ghosts are powerful and they give power to the person who believes in them. But they don't really exist, and the power is really in the person herself, she just does not know it. See how I change the meaning of the ghost also? In my explanation the frightening ghost is only frightening because it is strange, in the same way that a young child is afraid of cars or dogs, but when you understand the ghost, you will realize that it is a good thing, because it is actually your own power become manifest.
In short: make the ghost something positive, something positive about the mother and something positive in itself for the cild, and then integrate it into the childs self-image.
Also, try and understand the function of the fear in the present circumstances. The child has lost (?) its original family and is forced into a new situation. You are stangers to her, but she must trust you, because if she doesn't, she will be completely alone. In fact she is like a ghost herself, in a way, in danger of fading away, trying to grab onto the living.
An afterthought on frightening images.
Mostly we are afraid of the unknown. Frankenstein's Monster is most frightening as long as we don't see it. As soon as it actually appears, it becomes rather harmless and humanlike.
When you have a nightmare, you are awakening to get away from it. You are afraid to fall back asleep, because you have to face the nightmare again. But if you actually force youself to stay inside the dream (something you can do in the half-sleep state before you become fully awake), the nightmare will close in on you only to prove itself completely harmless. Nightmares that I have faced in this way (by not waking up fully and simply stopping to "run") have often turned into a reassuring and comforting emotion.
Try and help the girl to "meet" her ghost, then it will disappear, because it will transform into something benign.
