We demonstrate a GHz U-band fiber laser harnessing soliton self-frequency shift (SSFS). The seed source is a passively harmonic mode-locked (HML) fiber laser based on carbon nanotubes (CNTs) polymer film. By adjusting the pump power and polarization controller (PC) appropriately, the repetition rate can be tuned up to 1.104 GHz with a side mode suppression ratio (SMSR) of 42.5 dB. The seed source with different repetition rates can be frequency shifted from C-band to U-band in a piece of dispersion-shifted fiber (DSF) after amplification. We selected three repetition rates (457.8 MHz, 700.1 MHz, and 1104.1 MHz) and systematically studied the impact of different repetition rates on SSFS. The broad tuning range of wavelengths covering the entire U-band can be obtained by adjusting the pump power of the main amplifier. To the best of our knowledge, the demonstrated laser is the first U-band femtosecond fiber laser based on passively HML with a repetition rate over 1 GHz.