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At radioactive ion beam (RIB) facilities, ions of short-lived radionuclides are cooled and bunched in buffer-gas-filled Paul traps to improve the ion-beam quality for subsequent experiments. To deliver even colder ions, beneficial to RIB experiments' sensitivity or accuracy, we employ Doppler and sympathetic cooling in a Paul trap cooler-buncher. The improved emittance of Mg^{+}, K^{+}, and O_{2}^{+} ion beams is demonstrated by a reduced time-of-flight spread of the extracted ion bunches with respect to room-temperature buffer-gas cooling. Cooling externally-produced hot ions with energies of at least 7 eV down to a few Kelvin is achieved in a timescale of O(100 ms) by combining a low-pressure helium background gas with laser cooling. This is sufficiently short to cool short-lived radioactive ions. As an example of this technique's use for RIB research, the mass-resolving power in a multireflection time-of-flight mass spectrometer is shown to increase by up to a factor of 4.6 with respect to buffer-gas cooling. Simulations show good agreement with the experimental results and guide further improvements and applications. These results open a path to a significant emittance improvement and, thus, unprecedented ion-beam qualities at RIB facilities, achievable with standard equipment readily available. The same method provides opportunities for future high-precision experiments with radioactive cold trapped ions.