  Narrow-line Seyfert\,1 galaxies (NLS1s) with very small broad-line widths
(say, FWHM(\hb) $\la $ 1200\,\kms) represent the extreme type of Seyfert\,1
galaxies that have small black hole masses (\mbh) and/or high Eddington ratios
(\redd). Here we study the X-ray properties of a homogeneously and optically
selected sample of 13 such objects, termed as very narrow line Seyfert\,1
galaxies (VNLS1s), using archival \xmm\ data. It is found that the Fe K$\alpha$
emission line is at most weak in these objects. A soft X-ray excess is
ubiquitous, with the thermal temperatures falling within a strict range of
0.1--0.2\,keV. Our result highlights the puzzling independence of the thermal
temperature by extending the relations to even smaller FWHM(\hb), i.e., smaller
\mbh\ ($\sim 10^6$ \msun) and/or higher \redd. The excess emission can be
modeled by a range of viable models, though the disk reflection and
Comptonization models generally give somewhat better fits over the smeared
absorption and the $p$-free models. At the Eddington ratios around unity and
above, the X-ray spectral slopes in the 2--10\,keV band are systematically
flatter than the Risaliti et al.'s predictions of the relationship with \redd\
suggested previously. Short timescale (1--2 hours) X-ray variability is common,
which, together with the variability amplitude computed for some of the
objects, are supportive of the scenario that NLS1s are indeed AGN with
relatively small \mbh.
