We present a systematic analysis of the X-ray spectral properties of a sample of 22 'narrow-line' Seyfert 1 galaxies for which data are available from the ASCA public archive. Many of these sources, which were selected on the basis of their relatively narrow H beta linewidth (FWHM less than or equal to 2000 km s(-1)), show significant spectral complexity in the X-ray band. Their measured hard power-law continua have photon indices spanning the range 1.6-2.5 with a mean of 2.1, which is only slightly steeper than the norm for 'broad-line' Seyfert 1s. All but four of the sources exhibit a soft excess, which can be modelled as blackbody emission (T-bb approximate to 100-300 eV) superposed on the underlying power law. This soft component is often so strong that, even in the relatively hard bandpass of ASCA, it contains a significant fraction, if not the bulk, of the X-ray luminosity, apparently ruling out models in which the soft excess is produced entirely through reprocessing of the hard continuum. Most notably, six of the 22 objects show evidence for a broad absorption feature centred in the energy range 1.1-1.4keV, which could be the signature of resonance absorption in highly ionized material. A further three sources exhibit 'warm absorption' edges in the 0.7-0.9keV bandpass. Remarkably, all nine 'absorbed' sources have H beta linewidths below 1000 km s(-1), which is less than the median value for the sample taken as a whole. This tendency for very narrow linewidths to correlate with the presence of ionized absorption features in the soft X-ray spectra of NLS1s, if confirmed in larger samples, may provide a further clue in the puzzle of active galactic nuclei.