Typically, we recommend stain as opposed to paint for decks, because stain is thinner and penetrates deeper into the surface. Therefore, it does not wear off as quickly as paint. In your case however multiple layers of paint may be required to prevent slivers. Porch paint, typically provides thick coverage. The slivers are of serious concern because the deck is pressure treated. The chemical commonly used in pressure treating is copper arsenate salts. It is these salts that give the wood a greenish tint when it is new. Concerns have been raised about the toxicity of these chemicals, and some park and schoolyard play structures have been removed as a safety precaution, because there is concern that the chemicals may be carcinogenic. One or two thick coats of paint, or indoor/outdoor carpeting, or replacement of the deck boards with cedar are your options. In the meantime, you should probably keep your daughter off the deck.