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We demonstrate a wavelength-tunable all-optical flip-flop (AOFF) based on a multimode-interference (MMI) bistable laser diode (BLD) with distributed Bragg reflectors (DBRs). The wavelength tuning can be achieved by the band-filling effect and the free-carrier plasma effect with carrier injections to the DBRs. The tuning range of 3.1 nm will enable flexible and reconfigurable design of future photonic integrated circuits (PICs). Since the DBR-MMI-BLD does not require cleaved facets for its operation, multiple AOFFs can be monolithically integrated on a single PIC. In addition, the device has a large operable bandwidth of wider than 34 nm, even when we tune the lasing wavelength. A dynamic flip-flop operation is also demonstrated with rising and falling times of faster than 280 and 228 ps, respectively. No waveform distortion is observed, even when we tune the lasing wavelength.