  Aims. We aim at studying the near infrared linear polarization signal of
rapidly rotating ultracool dwarfs with spectral types ranging from M7 through
T2 and projected rotational velocities vsini >= 30 km s^{-1}. All these dwarfs
are believed to have dusty atmospheres and oblate shapes, an appropriate
scenario to produce measurable linear polarization of the continuum light.\\
Methods. Linear polarimetric images were collected in the J-band for a sample
of 18 fast-rotating ultracool dwarfs, five of which were also observed in the
Z-band using the LIRIS spectrograph on the Cassegrain focus of the 4.2-m
William Herschel Telescope. The measured median uncertainty in the linear
polarization degree is +/-0.13% for our sample, which allowed us to detect
polarization signatures above ~0.39% with a confidence of >=3\sigma.\\ Results.
About 40+/-15% of the sample is linearly polarized in the Z- and J-bands. All
positive detections have linear polarization degrees ranging from 0.4% to 0.8%
in both filters independently of spectral type and spectroscopic rotational
velocity. However, simple statistics point at the fastest rotators (vsini >=60
km s^{-1}) having a larger fraction of positive detections and a larger
averaged linear polarization degree than the moderately rotating dwarfs
(vsini=30--60 km s^{-1}). Our data suggest little linear polarimetric
variability on short time scales (i.e., observations separated by a few ten
rotation periods), and significant variability on long time scales (i.e.,
hundred to thousand rotation cycles), supporting the presence of "long-term
weather" in ultracool dwarf atmospheres.
