CBC-MAC based on PRF
The CBC-MAC construction indeed can use a PRF instead of PRP. It is now based on PRP due to historical reasons: the blockciphers used for CBC-MAC were based on permutations.

From the security point of view there will be no difference: the security proof for the CBC-MAC first converts PRP to PRF (which is indistinguishable up to $2^{n/2}$ queries) and then derives an actual upper bound on the adversary's advantage.