  SMSS J160540.18$-$144323.1 is the carbon-enhanced metal-poor (CEMP) star with
the lowest iron abundance ever measured, [Fe/H]=-6.2, which was first reported
with the SkyMapper telescope. The carbon abundance is A(C)~6.1 in the low-C
band, as the majority of the stars in this metallicity range. Yet, constraining
the isotopic ratio of key species, such as carbon, sheds light on the
properties and origin of these elusive stars. We performed high-resolution
observations of SMSS1605$-$1443 with the ESPRESSO spectrograph to look for
variations in the radial velocity ($v_{rad}$) with time. These data have been
combined with older MIKE and UVES archival observations to enlarge the temporal
baseline. The $^{12}$C/$^{13}$C isotopic ratio is also studied to explore the
possibility of mass transfer from a binary companion. A cross-correlation
function against a natural template was applied to detect $v_{rad}$ variability
and a spectral synthesis technique was used to derive $^{12}$C/$^{13}$C in the
stellar atmosphere. We confirm previous indications of binarity in
SMSS1605$-$1443 and measured a lower limit $^{12}$C/$^{13}$C$>60$ at more than
a 3$\sigma$ confidence level, proving that this system is chemically unmixed
and that no mass transfer from the unseen companion has happened so far. Thus,
we confirm the CEMP-no nature of SMSS1605$-$1443 and show that the pristine
chemical composition of the cloud from which it formed is currently imprinted
in its stellar atmosphere free of contamination.
