The paper Solving the Binary Linear Programming Model in Polynomial Time claims that Binary Integer Linear Programming is in P. However, it seems that no subsequent literature in the mainstream has done any further study on this. I am a bit doubtful regarding the correctness of the claim made in that paper. Therefore, I have put this question here.
Often called Binary Integer Programming (BIP). Wikipedia:
Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.
Here is a list of those 21 Karp problems.
You can also find the claim that BIP $\in$ NPC in many class notes, e.g., this set.