We use spectra and maps of NH2D, ND2H, and ND3, obtained with the CSO, IRAM 30 m and Arecibo telescopes, to study deuteration processes in dense cores. The data include the first detection of the hyperfine structure in ND2H. The emission of NH2D and ND3 does not seem to peak at the positions of the embedded protostars, but instead at offset positions, where outflow interactions may occur. A constant ammonia fractionation ratio in star-forming regions is generally assumed to be consistent with an origin on dust grains. However, in the pre-stellar cores studied here, the fractionation varies significantly when going from NH3 to ND3. We present a steady state model of the gas-phase chemistry for these sources, which includes passive depletion onto dust grains and multiply saturated deuterated species up to five deuterium atoms (e.g. CD5^+). The observed column density ratios of all four ammonia isotopologues are reproduced within a factor of 3 for a gas temperature of 10 K. We also predict that deuterium fractionation remains significant at temperatures up to about 20 K. ND and NHD, which have rotational transitions in the submillimeter domain are predicted to be abundant.