We report on the realization and operation of a fast ion beam trap of the linear electrostatic type employing liquid helium cooling to reach extremely low blackbody radiation temperature and residual gas density and, hence, long storage times of more than 5 min which are unprecedented for keV ion beams. Inside a beam pipe that can be cooled to temperatures <15 K, with 1.8 K reached in some locations, an ion beam pulse can be stored at kinetic energies of 2-20 keV between two electrostatic mirrors. Along with an overview of the cryogenic trap design, we present a measurement of the residual gas density inside the trap resulting in only 2 x 10(3) cm(-3), which for a room temperature environment corresponds to a pressure in the 10(-14) mbar range. The device, called the cryogenic trap for fast ion beams, is now being used to investigate molecules and clusters at low temperatures, but has also served as a design prototype for the cryogenic heavy-ion storage ring currently under construction at the Max-Planck Institute for Nuclear Physics.