The Gemara in Masechet Shabbat (48) addresses the question of whether it is permissible to fill a pillow with feathers or other material on Shabbat. If the pillow had not been previously filled, the Gemara establishes, then filling it on Shabbat would constitute "Makke Ba'patish" completing the process of making a functional item, one of the thirty-nine categories of Shabbat prohibitions. It appears from the Gemara that this would be forbidden on the level of Torah law ("Mi'de'orayta"). However, if the pillow was previously stuffed and the material had come out of the pillow, then returning the material inside the pillow would be permissible. (Of course, this assumes that one does not stitch the pillow, which would certainly be forbidden on Shabbat.) This Halacha is codified by the Shulhan Aruch (Orah Haim 340:8).
Rabbi Gedalya Felder (Toronto, 20th century), in his work Yesode Yeshurun, extends the Gemara's ruling to the contemporary issue of inflating balls on Shabbat. He rules that if the ball was already inflated before Shabbat, and subsequently became deflated, Halacha allows inflating it anew on Shabbat. If, however, the ball had not been inflated before Shabbat, then inflating it on Shabbat would be forbidden due to the prohibition of "Makke Be'patish," as discussed above.
Rabbi Shlomo Zalman Auerbach (Israel, 1910-1995), by contrast, as cited in the work "Shemirat Shabbat Ke'hilchata" (16:8), disagrees. In his view, one cannot compare the two cases of filling a pillow with feathers and filling a ball with air, and thus he maintains that Halacha allows inflating a ball even if it had not been previously inflated.
It should be noted that according to Sephardic practice, balls are in any event considered Mukse and may not be handled on Shabbat. For Sepharadim, then, this Halacha applies only in the case of a child, regarding whom Halacha is more lenient and allows handling and paying with a ball on Shabbat. Children are thus permitted to inflate a ball with which they wish to play on Shabbat, but adults should not inflate a ball or even handle it at all Shabbat.
Summary: It is permissible to inflate a ball on Shabbat. However, Sephardic custom forbids handling a ball in any event on Shabbat, except for children.
