A passively mode-locked Yb3+-doped fiber laser with a fundamental repetition rate of 5 GHz and wavelength tunable performance is demonstrated. A piece of heavily Yb3+-doped phosphate fiber with a high net gain coefficient of 5.7 dB/cm, in conjunction with a fiber mirror by directly coating the SiO2/Ta2O5 dielectric films on a fiber ferrule is exploited for shortening the laser cavity to 2 cm. The mode-locked oscillator has a peak wavelength of 1058.7 nm, pulse duration of 2.6 ps, and the repetition rate signal has a high signal-to-noise ratio of 90 dB. Moreover, the wavelength of the oscillator is found to be continuously tuned from 1056.7 to 1060.9 nm by increasing the temperature of the laser cavity. Simultaneously, the repetition rate correspondingly decreases from 4.945874 to 4.945496 GHz. Furthermore, the long-term stability of the mode-locked operation in the ultrashort laser cavity is realized by exploiting temperature controls. This is, to the best of our knowledge, the highest fundamental pulse repetition rate for 1-μm mode-locked fiber lasers.