If a variable length CBC-MAC is used with a block cipher (a.k.a. pseudorandom permutation), does there exist a block cipher such that implementation of this CBC-MAC may be used as a cryptographic hash function, provided that k is fixed?
No, it cannot; you can find a preimage attack on it.
CBC-MAC on a two-block message $(M_0, M_1)$ (after padding) is defined as $E_k( M_1 \oplus E_k( M_0 ))$. To find a preimage for a value $S$, you select $M_1$ with a valid padding pattern (because of the padding, we cannot select this block arbitrarily), and compute $M_0 = D_k(M_1 \oplus D_k(S))$; it is easy to see that the padding message $(M_0, M_1)$ hashes to the value $S$.
And, even if you cannot easily compute $D_k$, it is still easy to find second preimages...