Very flexible feet (but it's not always accompanied by a very flexible body!) are generally speaking, very weak in strength. They need to be 'tamed' into a reasonable, strong pointe.
What has been explained before is good. There is a need to build the strength to be able to push through the floor, rather than let gravity push the foot beyond the required arch. Usually, those feet are lovely looking, but for the student, it's a lot of hard work to catch up on the strength issue. An elastic (it has to be wide, and super resistant) across the foot helps 'tame' the foot a bit (it will prevent the foot from going 'over' the shoe) but it will not really train the foot to achieve this feeling without the elastic (or at least not as fast as with other options).
Training the foot itself, little by little is the answer. The use of high vamp (although not necessarily good for every feet) and strong shank (for support) is usually necessary, but the work needs mostly to be done on flats. I usually recommend not to use split sole on flats (because there's virtually no work of the foot done) so, in any case, full sole shoes, and ideally, soft pointes or deshanked, 1/2 pointes are good to build the strength within the foot.