From
https://dil.ie/21886"acc. to Pedersen Vgl. Gr., i 112 a derivative of fid `wood,' orig. an adj. = ` wild ,' cf. W. gwydd `wild' and Lat. silvaticus < silva, a sense preserved in compds., but early applied to wild animals in particular, and used as subst."
The deer meaning could be included in the entry on BTN.