The hyperfine structures of the $2,^3!S_1$ 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 [href{https://link.aps.org/doi/10.1103/PhysRevLett.111.243001}{Phys. Rev. Lett. {bf 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 [href{https://link.aps.org/doi/10.1103/PhysRevA.78.012513}{Phys. Rev. A {bf 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 $2,^3!P_J$ states of $^{6,7}$Li$^+$, where an order of magnitude improvement over the previous theory has been achieved for $^7$Li$^+$.
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