There are a few negative consequences of the use of antiperspirants, although your daughter may consider them to be, um, less consequential than slipping off the beam or vault table and crashing due to excessive perspiration. Formulations that contain alcohol (including Drysol) may dry the skin, so a gymnast would have to devote increased attention to avoiding rips, perhaps through regular use of moisturizers and also other skin care. Many antiperspirants should be applied to dry skin only (preferably at bedtime), since the ingredients can react with water/sweat to produce acid that irritates the skin, and washed off in the morning. In addition to Drysol, commercial products with high concentrations of the aluminum salts are available (e.g. Secret Clinical Strength). Like Drysol, these products don't eliminate sweating, but only reduce it, so finding the right balance of comfort, convenience, and efficacy for your daughter should be important.
A tapwater iontophoresis system is available for home use; this could control sweating for weeks. Your daughter would have to be highly motivated, since this requires repeated and somewhat time-consuming treatments to achieve such results. Several studies suggest that this should benefit about 85% of patients. This may be worth a try if she isn't satisfied with the results from the use of antiperspirants.
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You might consider treating her feet with a solution of glutaraldehyde in water (you should be able to get this from a pharmacist after consulting a dermatologist). The downside of this treatment is that it causes some discoloration of the skin (the treated area turns yellow/brown--that's why it is usually used only on the soles of the feet, which in nongymnasts are usually much less visible than the palms of the hands); a stronger (10%) solution works well, but discolors more than a weaker (2%) solution that is less effective.
The only permanent solution requires (endoscopic) surgery to cut nerves, and—in addition to the usual possible problems associated with any surgery—this procedure often leads to compensatory sweating elsewhere.