I suspect the answer to this question will somewhat depend on your definition of an encryption scheme. In particular, can the key have an infinite number of bits, and can the encryption function Enc be stateful (so that it can make sure to never use the same key bits twice)?
If the answer to both questions is yes, then I believe it should be possible to define a one-time-pad-like scheme that is perfectly CPA-secure. The basic idea is that the key should consist of (or can be bijectively mapped into) an infinite number of infinitely long bit streams, and the encryption algorithm stores a message counter. To encrypt a message, it first increments the counter, extracts the bit stream indicated by the counter from the key, takes the prefix of this bit stream equal to the message in length, and XORs it with the message. Finally, it prepends the current value of the counter to the message (in some manner that allows it to be unambiguously extracted by the decryption algorithm) and returns the result. If I'm not mistaken, a scheme like this should provide perfect secrecy for an unlimited number of messages, and thus be perfectly CPA-secure even against arbitrarily powerful adversaries.
On the other hand, if the encryption algorithm is not stateful (and so cannot remember which key streams it has already used), then there's a non-zero change that it will use the same key material to encrypt two messages. And if the key material itself is finite, then this must eventually happen after sufficiently many encryptions, whether the encryption algorithm is stateful or not. Either way, an attacker with unlimited oracle access can exploit this by repeatedly querying the oracle for encryptions of their challenge messages until the result matches the challenge ciphertext.
In fact, for non-stateful encryption algorithms, the attacker doesn't even need unlimited oracle access to have a non-zero advantage, since there's always a non-zero probability that even a single query will just happen to yield a match. We can make this probability negligible, but not zero. On the other hand, a stateful encryption algorithm could be designed to avoid reusing a keystream for any given number of encryptions, but only if the key was long enough to permit this. In particular, asymptotically, a super-polynomial number of queries before a keystream is reused would seem to require a super-polynomial key length (and thus, presumably, a super-polynomial running time for the key generator).