  We present a 1.2 mm (250 GHz) map obtained with MAMBO on the IRAM 30m
telescope of the central 25 arcmin^2 of the proto-cluster surrounding the z=4.1
radio galaxy TN J1338-1942. The map reaches a 1sigma sensitivity of 0.6 mJy in
the central area, increasing to 1.2 mJy at the edges. We detect 10 candidate mm
sources, of which 8 are also detected in a deep VLA 1.4 GHz map and/or a VLT
R-band image. Three sources have a flux density S_{1.2 mm}>4.0 mJy,
representing a 7sigma overdensity compared to random field surveys, which
predict only 1 such source in our map area. We obtained SCUBA/JCMT 850 um and
450 um photometry of six radio/optically identified MAMBO sources, confirming 5
of them with S/N>4. Radio-to-mm and mm-to-submm redshift estimators cannot put
strong constraints on the redshifts of these MAMBO sources, but 9 of them are
consistent within the uncertainties (mean Delta z=+2.6) with z=4.1. One faint
MAMBO source is possibly identified with an extremely red object (R-K=6.1) at a
likely spectroscopic redshift z=1.18. The four brightest MAMBO sources are all
located north of the radio galaxy, while the densest area of companion Ly-alpha
excess and Lyman break galaxies is to the southeast. None of the 14
spectroscopically confirmed Ly-alpha emitters in the MAMBO field are detected
at 1.2 mm; their average 1.2 mm flux density is <S_{\rm 1.2mm}>=0.25+-0.24 mJy.
If the mm sources lie at z=4.1, none of them show excess Ly-alpha emission in
our narrow-band images. Both populations thus show no apparent overlap,
possibly due to dust quenching the Ly-alpha emission. If the mm sources are
part of the proto-cluster, our results suggest that galaxies with star
formation rates of a few 1000 M_{Sun} yr^{-1} could be spread throughout the
proto-cluster over projected scales of at least 2 Mpc.
