What are the properties of a function based on multiple iterations of a pseudo-random permutation?
It is neither, because you're only talking about a fixed function—it's not even the right type of object to be a PRF or PRP.  Keccak-f[1600] is not a pseudorandom permutation family itself; it is a single fixed permutation.  A PRF or a PRP is a family of functions indexed by a key, such that a uniform distribution on keys induces an apparently uniform distribution on functions or permutations in some space.