The hyperfine structures of the 2 3 S1 states of the 6 Li + and 7 Li + ions are investigated theoretically to extract the Zemach radii of the 6 Li and 7 Li nuclei by comparing with precision measurements. The obtained Zemach radii are larger than the previous values of Puchalski and Pachucki [Phys. Rev. Lett. 111, 243001 (2013)] and disagree with them by about 1.5 and 2.2 standard deviations for 6 Li and 7 Li, respectively. Furthermore, our Zemach radius of 6 Li differs significantly from the nuclear physics value, derived from the nuclear charge and magnetic radii [Phys. Rev. A 78, 012513 (2008)], by more than 6 sigma, indicating an anomalous nuclear structure for 6 Li. The conclusion that the Zemach radius of 7 Li is about 40% larger than that of 6 Li is confirmed. The obtained Zemach radii are used to calculate the hyperfine splittings of the 2 3 PJ states of 6,7 Li + , where an order of magnitude improvement over the previous theory has been achieved for 7 Li + .