  Showing 1.4 GHz flux densities in the range of a few to a few tens of mJy,
infrared-faint radio sources (IFRS) are a type of galaxy characterised by faint
or absent near-infrared counterparts and consequently extreme radio-to-infrared
flux density ratios up to several thousand. Recent studies showed that IFRS are
radio-loud active galactic nuclei (AGNs) at redshifts >=2. This work explores
the far-infrared emission of IFRS, providing crucial information on the star
forming and AGN activity of IFRS and on the potential link between IFRS and
high-redshift radio galaxies (HzRGs). A sample of six IFRS was observed with
the Herschel Space Observatory between 100 um and 500 um. Using these results,
we constrained the nature of IFRS by modelling their broad-band spectral energy
distribution (SED). Furthermore, we set an upper limit on their infrared SED
and decomposed their emission into contributions from an AGN and from star
forming activity. All six observed IFRS were undetected in all five Herschel
far-infrared channels (stacking limits: sigma = 0.74 mJy at 100 um, sigma =
3.45 mJy at 500 um). Based on our SED modelling, we find that the IFRS analysed
in this work can only be explained by objects that fulfil the selection
criteria of HzRGs. More precisely, IFRS could be (a) known HzRGs at very high
redshifts (z >= 10.5), (b) low-luminosity siblings of HzRGs with additional
dust obscuration at lower redshifts, (c) scaled or unscaled versions of Cygnus
A at any redshift, and (d) scaled and dust-obscured radio-loud quasars or
compact steep spectrum sources. We estimated upper limits on the infrared
luminosity, the black hole accretion rate, and the star formation rate of IFRS,
which all agreed with corresponding numbers of HzRGs.
