Why are lithides not known?
One barrier is the greater tendency of lithium to form partially covalent bonds than heavier alkali metals.  This is a good thing for organic chemists, since this stabilizes organolithium compounds and enhances the solubility of lithium chloride as a chloride ion source (see this answer).  But it's not so hot for forming lithides.  Covalent bonding to more electronegative elements, meaning to practically all other elements in this case, oxidizes the intended lithide ion.  In effect the cryptand in an intended lithide is converted into one of those organolithium compounds.

Linear Christmas, in the comments to the question, hints at a way to beat this tendency.  Two of the references given there, https://doi.org/10.1021/ja056314+ and https://doi.org/10.1002/cphc.201200805, involve theoretical investigations of lithides built with pyrrole-based cryptands.  Such cryptands resist covalent bonding to lithide (or other nucleophilic) anions because this would destroy the aromatic of the ring and require electrons to overlap into a high-lying, strongly antibonding orbital.  These compounds, or counterparts with heavier alkalide ions, draw interest for their expected nonlinear optical (NLO) properties, making them synthetic targets.  So, stay tuned.