A narrow-linewidth laser operating at the telecommunications band combined with both fast and wide-band tuning features will have promising applications. Here we demonstrate a single-mode (both transverse and longitudinal mode) continuous microlaser around 1535 nm based on a fiber Fabry–Pérot microcavity, which achieves wide-band tuning without mode hopping to the 1.3 THz range and fast tuning rate to 60 kHz and yields a frequency scan rate of 1.6×1017Hz/s. Moreover, the linewidth of the laser is measured as narrow as 3.1 MHz. As the microlaser combines all these features into one fiber component, it can serve as the seed laser for versatile applications in optical communication, sensing, frequency-modulated continuous-wave radar, and high-resolution imaging.