Encryption schemes and pseudorandom permutations Can an encryption scheme (deterministic or not) be viewed as a pseudorandom permutation taking as input a message and returning a value of the same bit-encoding length?
The term "pseudorandom permutation" is normally used for a family of (computable) bijections $f_k : M \to M$, where $M$ is some finite set, and usually $M = \{0,1\}^n$. (Usually we also have a computable inverse for each $f_k$, and some security properties.)

If we stretch the meaning a bit, we can expand this to any finite set $M$, like "the set of all byte strings shorter than $2^{100}$ bytes" (i.e. more than anything you would ever want to encrypt. But when using this set, all common encryption schemes are, while permutations, not pseudorandom, because we can guess from the length of the output to the length of the input.

If we instead use as $M$ a set like "the set of all byte strings of length 1 KB", any good deterministic encryption scheme restricted to $M$ indeed should fulfill the properties of a random permutation on $M$.

A non-deterministic encryption scheme first will have to expand the ciphertext (e.g. by writing the IV into it), and secondly is not a permutation (a permutation is deterministic).

If we can extract the randomness part and use it as a separate input to a deterministic function (or, as a chooser for a family member), we can treat the encryption itself just like before.

But normally the term "random permutation" is used only for the actual building blocks of such encryption schemes, also named "block cipher".