  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$^+$.
