Do non-classical carbanions exist?
Interesting question. It is much less studied and reported on than the case of non-classical carbocations, but I did find a few papers. Brown and Occolowitz (Ref.1) reported that deuterated bicyclo[3.2.1]octa-2,6-diene 1b, below, undergoes base-catalysed de-deuteration (potassium tert-butoxide in DMSO) to give 1a much faster (ca. $3 \times 10^4$) than the bicyclo[3.2.1]octa-2-ene counterpart, i.e. 2b $\rightarrow$ 2a. They suggested this is evidence for the intermediacy of a 6&pi;-electron, delocalised (non-classical) carbanion 3.



Much more recently, Brown reviewed and provided additional DFT computational evidence for stabilisation and charge delocalisation in these species (Ref.2).

References:


J. M. Brown, J. L. Occolowitz, “A non-classical carbanion,” Chem. Commun. (London) 1965, (16), 376-377 (https://doi.org/10.1039/C19650000376).
John M. Brown, “Origins of Stabilization and Evidence for Charge Delocalization in the Bicyclo[3.2.1]octadienyl Anion and Related Species,” Australian Journal of Chemistry 2014, 67(9), 1296-1300 (https://doi.org/10.1071/CH14244).