Context : A number of cool magnetic Ap stars show a prominent feature at \lambda 6708 Å .
Its identification with Li i remains controversial due to a poor knowledge of the spectra of rare-earth elements that are strongly enhanced in peculiar stars and can potentially provide an alternative identification .

Aims : We suggest to investigate the 6708 Å line in Ap stars with strong magnetic fields .
In these objects the magnetic broadening and splitting provides an additional powerful criterium for line identification , allowing to use the whole line profile instead of a mere coincidence of the observed and predicted wavelength .

Methods : Due to a small separation of the Li i doublet components , their magnetic splitting pattern deviates from the one expected for the Zeeman effect even in relatively weak fields .
We carry out detailed calculations of the transition between the Zeeman and Paschen-Back regimes in the magnetic splitting of the Li i line and compute polarized synthetic spectra for the range of field strength expected in Ap stars .
Theoretical spectral synthesis is compared with the high-resolution observations of cool Ap stars HD 116114 , HD 166473 and HD 154708 , which have a mean field strength of 6.4 , 8.6 and 24.5 kG , respectively , and show a strong 6708 Å line .

Results : High-resolution spectra for the 6708 Å region are analysed for 17 magnetic Ap stars .
The presence of the 6708 Å line is confirmed in 9 stars and reported for the first time in 6 stars .
The strength of the Li i doublet does not correlate with absorption features of any other element .
The stars HD 75445 and HD 201601 provide an extreme example of the two objects which are dissimilar with respect to the 6708 Å line but very close in the atmospheric parameters and abundances of other elements .
We demonstrate that the observed profiles of the 6708 Å line in the strong field stars HD 116114 , HD 166473 and HD 154708 correspond rather well to the theoretical calculations assuming the Li i identification .
Inclusion of the Paschen-Back effect improves the agreement with observations , especially for HD 154708 .

Conclusions : Results of our study confirm the Li i identification proposed for the 6708 Å line in cool Ap stars .