Maybe, the paper entitled"Extraction of lithium from the dead sea" is helpful to  lithium via solvent extraction. Good luck!

http://www.sciencedirect.com/science/article/pii/0304386X8190044X

Which is the final product? do you need the water or the lithium? And is it lithium as an element you want to obtain or an lithium based compound? You could first of all remove the water by destillation, then you will have the salts in a pure form. If you want to have lithium as a metal, try to reduce it electrolytically. by applying the right potentials you can easily obtain the pure lithium separated from strontium or other species... If it has to be in an extraction reaction the most appropirate way will shurely be the application of 12crown4 in a toluene phase for example. but keep in mind, that you need two equivalents of 12crown4 to chelate the lithium+ effectively...

Another way is to use cation exchange chromatography. perhaps there is something at the market that only captures lithium...? take a look here: http://pubs.acs.org/doi/abs/10.1021/j150537a002?journalCode=jpchax

Ion exchange in a column is not an option?

For extracting lithium from water, simple solvent extraction is not the answer. Typically natural water will also contain sodium, potassium, calcium and smaller amounts of other ions. I only know one selective extractant for lithium, namely a 14-crown-4 ether derivatized with nitrophenol or azophenol.

See Chemistry Letters 1985, 14, 1239-40

https://www.jstage.jst.go.jp/article/cl1972/14/8/14_8_1239/_pdf

Possibly ion-exchange chromatography could be a better choice.

Maybe, the paper entitled"Extraction of lithium from the dead sea" is helpful to  lithium via solvent extraction. Good luck!

http://www.sciencedirect.com/science/article/pii/0304386X8190044X

Along the previous lines, the article in Hydrometallurgy 1991 (27, 317-325) by Bukowsky et al. should also be helpful.

The recovery of pure lithium chloride from

"brines" containing higher contents of calcium

chloride and magnesium chloride

http://www.sciencedirect.com/science/article/pii/0304386X9190056R/pdf?md5=c1f148f4d925820e9a2927bea9a0b1aa&pid=1-s2.0-0304386X9190056R-main.pdf

In any case, concentration is an indispensable initial step to yield a "brine" from which lithium can then be extracted and separated from other alkali and alkaline earth ions.