Unforgeable CPA-secure private-key encryption scheme that is not CCA-secure
Take an authenticated (unforgeable and CCA-secure) encryption scheme, such as the one constructed in the book using encrypt-then-authenticate with a CPA-secure encryption and a strongly secure MAC. 

Define a new scheme by appending to the ciphertext a random string. 

This new scheme is still CPA secure and unforgeable. But it's not CCA secure because the adversary can change the random appendix and thus ask the oracle for decryption and win the CCA game.