  The hydrogen Paschen lines are known activity indicators, but studies of them
in M~dwarfs during quiescence are as rare as their reports in flare studies.
This situation is mostly caused by a lack of observations, owing to their
location in the near-infrared regime, which is covered by few high-resolution
spectrographs. We study the Pa$\beta$ line, using a sample of 360 M~dwarfs
observed by the CARMENES spectrograph. Descending the spectral sequence of
inactive M~stars in quiescence, we find the Pa$\beta$ line to get shallower
until about spectral type M3.5 V, after which a slight re-deepening is
observed. Looking at the whole sample, for stars with H$\alpha$ in absorption,
we find a loose anti-correlation between the (median) pseudo-equivalent widths
(pEWs) of H$\alpha$ and Pa$\beta$ for stars of similar effective temperature.
Looking instead at time series of individual stars, we often find correlation
between pEW(H$\alpha$) and pEW(Pa$\beta$) for stars with H$\alpha$ in emission
and an anti-correlation for stars with H$\alpha$ in absorption. Regarding
flaring activity, we report the automatic detection of 35 Paschen line flares
in 20 stars. Additionally we found visually six faint Paschen line flares in
these stars plus 16 faint Paschen line flares in another 12 stars. In strong
flares, Paschen lines can be observed up to Pa 14. Moreover, we find that
Paschen line emission is almost always coupled to symmetric H$\alpha$ line
broadening, which we ascribe to Stark broadening, indicating high pressure in
the chromosphere. Finally we report a few Pa$\beta$ line asymmetries for flares
that also exhibit strong H$\alpha$ line asymmetries.
