We present results of dust continuum and [C II]$\, 158\, {\rm \mu m}$ emission line observations of a remarkably UV luminous (MUV = -21.6) galaxy at z = 10.603: GN-z11. Using the Northern Extended Millimeter Array (NOEMA), observations have been carried out over multiple observing cycles. We achieved a high sensitivity resulting in a $\lambda _{\rm rest}=160\, {\rm \mu m}$ continuum $1\, \sigma$ sensitivity of $13.0\, \rm {\mu Jy \, beam}^{ -1}$ and a [C II] emission line $1\, \sigma$ sensitivity of $31\, \rm {mJy\, beam^{ -1}\, km \, s}^{ -1}$ using $50\, \rm {km \, s}^{ -1}$ binning with a $\sim 2\, {\rm arcsec}$ synthesized beam. Neither dust continuum nor [C II]$\, 158\, {\rm \mu m}$ line emission are detected at the expected frequency of ν[C II]$= 163.791\, \rm {GHz}$ and the sky location of GN-z11. The upper limits show that GN-z11 is neither luminous in LIR nor L[C II], with a dust mass $3\, \sigma$ limit of ${\rm log}\, (M_{\rm dust}/{\rm {\rm M}_{\odot }}) \, \lt\, 6.5-6.9$ and with a [C II] based molecular gas mass $3\, \sigma$ limit of log (Mmol, [C II]$/{\rm {\rm M}_{\odot }}) \, \lt \, 9.3$. Together with radiative transfer calculations, we also investigated the possible cause of the dust poor nature of the GN-z11 showed by the blue colour in the UV continuum of GN-z11 (βUV = -2.4), and found that ≳3 × deeper observations are crucial to study dust production at very high-redshift. Nevertheless, our observations show the crucial role of deep mm/submm observations of very high-redshift galaxies to constrain multiple phases in the interstellar medium.