Is it safe to assume Salsa20 to be a PRP? Often in security proofs a certain block cipher is assumed to be a pseudorandom permutation or PRP. I wonder if this goes for stream ciphers as well, and specifically for Salsa20.
If we limit ourselves to the scope of one Salsa20 block (64 bytes), being fed 64 bytes of plaintext data, is it safe to assume $Salsa20_{key,nonce}$ to be a (strong) PRP family?
I'd very much be interested in the reasoning why or why not, and whether this extends to other stream ciphers.
Salsa20 is a stream cipher based on a pseudorandom function, not a pseudorandom permutation.

For a fixed key $k$ and nonce $n$, the mapping $PRF^{S20}_{k,n}: \{0,1\}^{64} \to \{0,1\}^{512}$, which maps a "Stream position" to "keystream block", is supposed to be a pseudorandom function. It is not supposed to be injective (i.e. a permutation, even less since the input/output sizes are different), and there certainly is no easy way to produce a preimage, even knowing the key and nonce.

What you propose is the Salsa20 encryption function for the first "block" of plaintext:
$$\def\Enc{\operatorname{Enc}}\Enc^{S20}_{k, n} : \{0,1\}^{512} \to \{0,1\}^{512}$$
$$\Enc^{S20}_{k, n}(P) = P \oplus PRF^{S20}_{k,n}(0)$$

This is obviously bijective (it is its own inverse), i.e. a permutation, but at the same time, it is obviously not a (pseudo)random permutation. In addition to the self-inverse-property, it for example also has the two-time-pad property:

$$\Enc^{S20}_{k, n}(P_1) \oplus \Enc^{S20}_{k, n}(P_2) = P_1 \oplus P_2,$$

which no random permutation would have.