  The cosmic infrared background (CIB) traces star-forming galaxies throughout
cosmic history, with emission peaking at $z\sim1-2$. CIB anisotropies are
present at the far-infrared frequencies observed by cosmic microwave background
(CMB) experiments such as $\textit{Planck}$. These contain a lot of
astrophysical and cosmological information, but are hard to separate from the
dust emission in our own Milky Way galaxy, especially on large scales where the
Milky Way contamination severely dominates. This galactic component is often
cleaned using information from other galactic tracers such as neutral hydrogen
(HI). In this work we use HI data from the HI4PI survey to clean the 353, 545,
and 857 GHz $\textit{Planck}$ NPIPE single-frequency maps using a needlet
internal linear combination (NILC) method, with $\texttt{pyilc}$. This allows
us to preserve the CIB anisotropy information on $\textit{all}$ scales, while
reducing the variance sourced by the galactic contamination. We also create a
NILC CMB map from the $\textit{Planck}$ NPIPE data, to subtract a CMB template
from the 353 GHz map. Our resulting CIB maps are appropriate for
cross-correlation studies with cosmological tracers such as CMB lensing maps
down to very low $\ell$ ($\ell\sim10$), while achieving similar performance to
previous works on intermediate scales. The use of the NPIPE data additionally
allows us to achieve lower instrumental noise in the maps than in previous
works. We use our maps, in combination with the $\textit{Planck}$ NPIPE CMB
lensing reconstruction, to measure the CIB-CMB lensing cross correlation down
to $\ell\sim10$. We make various versions of our maps publicly available to the
community for further use in cross-correlation studies, along with a script
(and the intermediate data products required) to produce dust-cleaned CIB maps
on an arbitrary region of sky.
