We have measured the device length and temperature dependence of the intermediate frequency (IF) bandwidth and noise of hot-electron bolometer (HEB) mixers made from a high-T_c superconductor. Mixer devices with lengths (L) between 50 nm and 1 µm were fabricated from 25-35 nm thick YBCO films on MgO and sapphire substrates. Bandwidth measurements were made using signal and local oscillator (LO) frequencies in the range 1-20 GHz. At low operation temperatures the IF bandwidths were about 100 MHz and several hundred MHz for devices on MgO and sapphire, respectively. At higher operation temperatures, where self-heating disappeared and flux-flow effects define the shape of the I-V characteristic, the measured IF bandwidth increased significantly. The temperature and IF dependence of absolute conversion efficiencies determined from noise measurements are in good agreement with the bandwidth data. At 2.7 GHz LO frequency the single-side-band mixer noise temperature of a 50 nm long HEB on MgO was about 8000 K.