What do you do with a "child prodigy" ?
Same as any student. Try to challenge them and find as many opportunities for them as possible.
Personally, I would probably send them to a better teacher.
This response, for all its brevity, points as closely as any so far towards the best answer to m1469's question.
I should begin by declaring that I am neither a pianist nor a teacher, so in one sense I might seem to be even less well qualified to answer her question than those of you who are pianists and teachers but who have nevertheless yet to experience a child prodigy in your charge. I have, however, encountered this prodigy "problem" on a number of occasions and have had opportunites to observe and consider.
I think a lot will inevitably depend on the age of the child concerned; what one may do with a 5-year-old who has played for 2 years and developed the facilities expected of an averagely gifted student twice his/her age will likely be rather different to what one may do with an 8-year-old who has been playing for five years and, hand size considerations apart, can just about get their fingers around Chopin's Op. 10 (and yes, I have seen a case like that - and even another of an 11-year-old who could give a more than tolerable account of Liszt's Sonata and Alkan's Sonatine, albeit not from memory). A case such as the latter will have played quite a lot more music than the former; this fact, coupled with the greater age, will make it more likely that one could interest the student in a wider variety of challenging repertoire to explore.
I would also want to ensure that the child's listening experiences were as ample and as wide-ranging as possible, for, if they are, it will usually be somewhat easier to challenge him/her with a greater variety of repertoire; just as Busoni used to counsel pianists to practise the most difficult passage in a piece immediately next to the easiest, it's generally a good idea to put music of the English Elizabethan keyboard composers in front of such a child along with Bartók, Prokofiev and Shostakovich, as well as non-tonal music such as Schönberg's op. 19 and some of the easier pieces by Krenek, to avoid any risk that boredom may ensue as a result of exposure to an insufficient variety of harmonic and pianistic languages and techniques.
Get them also to listen to good performances of piano music by Ligeti, Xenakis, Boulez, Messiaen, etc. as well as Bach, Beethoven, Chopin et al.
Encourage them to begin to learn a second instrument if they are not already doing so.
Get them to play chamber music rather than working alone most of the time.
Get them to compose! - and encourage a certain amount of improvisation along with learning pieces from scores.
They won't necessarily like it all, of course, but at least they'll have been given plenty to think about!
I hope that this helps.
Best,
Alistair