Evidence is growing for a class of gamma-ray bursts ( GRBs ) characterized by an initial \sim 0.1 - 1 s spike of hard radiation followed , after a \sim 3 - 10 s lull in emission , by a softer period of extended emission lasting \sim 10 - 100 s. In a few well-studied cases , these “ short GRBs with extended emission ” show no evidence for a bright associated supernova ( SN ) . We propose that these events are produced by the formation and early evolution of a highly magnetized , rapidly rotating neutron star ( a “ proto-magnetar ” ) which is formed from the accretion-induced collapse ( AIC ) of a white dwarf ( WD ) , the merger and collapse of a WD-WD binary , or , perhaps , the merger of a double neutron star binary . The initial emission spike is powered by accretion onto the proto-magnetar from a small disk that is formed during the AIC or merger event . The extended emission is produced by a relativistic wind that extracts the rotational energy of the proto-magnetar on a timescale \sim 10 - 100 s. The \sim 10 s delay between the prompt and extended emission is the time required for the newly-formed proto-magnetar to cool sufficiently that the neutrino-heated wind from its surface becomes ultra-relativistic . Because a proto-magnetar ejects little or no ^ { 56 } Ni ( < 10 ^ { -3 } M _ { \sun } ) , these events should not produce a bright SN-like transient . We model the extended emission from GRB060614 using spin-down calculations of a cooling proto-magnetar , finding reasonable agreement with observations for a magnetar with an initial rotation period of \sim 1 ms and a surface dipole field of \sim 3 \times 10 ^ { 15 } G. If GRBs are indeed produced by AIC or WD-WD mergers , they should occur within a mixture of both early and late-type galaxies and should not produce strong gravitational wave emission . An additional consequence of our model is the existence of X-ray flashes unaccompanied by a bright SN and not associated with massive star formation .