The Pristine survey -- XXII. A serendipitous discovery of an extremely Li-rich very metal-poor giant and a new method of $^6$Li/$^7$Li isotope measurement

We report the serendipitous discovery of a very metal-poor (VMP) Li-rich giant star ($T_{\rm eff}$ = 4690$\pm$80 K, log g = 1.34$\pm$0.13, [Fe/H] = $-2.43\pm$0.07). We analyse the Li I 6103 and 6707 \r{A} lines accounting for departures from local thermodynamic equilibrium (NLTE) and correcting for 3D effects using literature data, which yields a lithium abundance $\log\varepsilon_{Li} = 3.42\pm0.07$. Comparing lithium abundances from the two lines, in 1D NLTE we measure the isotope ratio $^6$Li/$^7$Li = 1.64$^{+1.49}_{-1.08}$ %. When correcting for 3D effects, we detect the fragile $^6$Li isotope at $2$-sigma level and the ratio $^6$Li/$^7$Li = 5.65$^{+5.05}_{-2.51}$ %. To our knowledge, this is the first $^6$Li/$^7$Li measurement in an extremely Li-rich VMP star. The Cameron-Fowler mechanism, which is proposed to produce Li-rich stars, does not imply $^6$Li production and is therefore inconsistent with our measurement when applying 3D corrections. We also derive NLTE abundances for 16 elements, most of which show similar abundances to those found in VMP stars. Sodium is an exception: [Na/Fe]$_{\rm NLTE, 1D}$ = 0.07 $\pm 0.03$, which is 0.5 dex higher than what is typical for VMP stars. This star joins the sample of rare Li-rich VMP stars, and we offer a novel way to constrain the source of lithium in such stars through isotope ratio measurements.