We report the amplification of 250~GHz pulses as short as 260~ps without observation of pulse broadening using a photonic-band-gap circuit gyrotron traveling-wave-amplifier. The gyrotron amplifier operates with 38~dB of device gain and 8~GHz of instantaneous bandwidth. The operational bandwidth of the amplifier can be tuned over 16 GHz by adjusting the operating voltage of the electron beam and the magnetic field. The amplifier uses a 30~cm long photonic-band-gap interaction circuit to confine the desired TE$_{03}$-like operating mode while suppressing lower order modes which can result in undesired oscillations. The circuit gain is $>$55~dB for a beam voltage of 23~kV and a current of 700~mA. These results demonstrate the wide bandwidths and high gain achievable with gyrotron amplifiers. The amplification of picosecond pulses of variable lengths, 260-800~ps, shows good agreement with theory using the coupled dispersion relation and the gain-spectrum of the amplifier as measured with quasi-CW input pulses.
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