We discuss the concept of unextendible product basis (UPB) and generalized UPB for fermionic systems, using Slater determinants as an analogue of product states, in the antisymmetric subspace $\wedgeˆ N \bCˆM$. We construct an explicit example of generalized fermionic unextendible product basis (FUPB) of minimum cardinality $N(M-N)+1$ for any $N\ge2,M\ge4$. We also show that any bipartite antisymmetric space $\wedgeˆ 2 \bCˆM$ of codimension two is spanned by Slater determinants, and the spaces of higher codimension may not be spanned by Slater determinants. Furthermore, we construct an example of complex FUPB of $N=2,M=4$ with minimum cardinality 5. In contrast, we show that a real FUPB does not exist for $N=2,M=4$. Finally we provide a systematic construction for FUPBs of higher dimensions using FUPBs and UPBs of lower dimensions.