Context : Extremely metal-poor stars provide us with indirect information on the first generations of massive stars .
The TOPoS survey has been designed to increase the census of these stars and to provide a chemical inventory that is as detailed as possible .

Aims : Seven of the most iron-poor stars have been observed with the UVES spectrograph at the ESO VLT Kueyen 8.2 m telescope to refine their chemical composition .

Methods : We analysed the spectra based on 1D LTE model atmospheres , but also used 3D hydrodynamical simulations of stellar atmospheres .

Results : We measured carbon in six of the seven stars : all are carbon-enhanced and belong to the low-carbon band , defined in the TOPoS II paper .
We measured lithium ( A ( Li ) =1.9 ) in the most iron-poor star ( SDSS J1035+0641 , [ Fe/H ] < -5.2 ) .
We were also able to measure Li in three stars at [ Fe/H ] \sim - 4.0 , two of which lie on the Spite plateau .

We confirm that SDSS J1349+1407 is extremely rich in Mg , but not in Ca .

It is also very rich in Na .
Several of our stars are characterised by low \alpha -to-iron ratios .

Conclusions : The lack of high-carbon band stars at low metallicity can be understood in terms of evolutionary timescales of binary systems .
The detection of Li in SDSS J1035+0641 places a strong constraint on theories that aim at solving the cosmological lithium problem .
The Li abundance of the two warmer stars at [ Fe/H ] \sim - 4.0 places them on the Spite plateau , while the third , cooler star , lies below .

We argue that this suggests that the temperature at which Li depletion begins increases with decreasing [ Fe/H ] .
SDSS J1349+1407 may belong to a class of Mg-rich EMP stars .
We can not assess if there is a scatter in \alpha -to-iron ratios among the EMP stars or if there are several discrete populations .
However , the existence of stars with low \alpha -to-iron ratios is supported by our observations .