Splitting balanced incomplete block designs were first formulated by Ogata, Kurosawa, Stinson, and Saido recently in the investigation of authentication codes. This article investigates the existence of splitting balanced incomplete block designs, i.e., (v, 3k, λ)-splitting BIBDs; we give the spectrum of (v, 3 × 2, λ)-splitting BIBDs. As an application, we obtain an infinite class of 2-splitting A-codes.