Functional programming generally involves a lot of dynamic memory allocations (which are usually optimized in some way). Without some form of assisted memory management, this quickly becomes excruciating. C also lacks function closures, so important abstractions like function composition, partial application and higher order functions. It also lacks type parameterization and polymorphism.

Not everyone will consider all of these to be fundamental to functional programming, but I'm certain most people would require at least one. C simply doesn't provide powerful enough mechanisms for abstraction to enable any interesting parts of functional programming.

C++, Rust, D and Nim all do, however. If one ever felt the need to use them in a functional matter it would be possible, albeit unpleasant.

Additional reading: https://news.ycombinator.com/item?id=8609775