In his paper about RFSB Bernstein states that the compression function
$(m_1,\ m_2,\ m_3,\ ...\ ,\ m_n) \rightarrow c_1[m_1]\ \oplus\ c_2[m_2]\ \oplus c_3[m_3]\ \oplus\ ... \oplus\ c_n[m_n]$
is surpsingly collsion resistant. Reading this paper, I asked myself weather one can use ChaCha with increasing counter instead of arrays to reach nonlinearity. To do so, one would use the alternative initial matrix
k k k k
m m m m
m m m m
c c c c
where $k$ are the constants provided by Bernstein, $m$ is the message (which takes the place of the key) and $c$ is the counter which was expanded to 128 bit since there is no need for a nonce in this scheme. The usual ChaCha algorithm is applied to this matrix.
Obviously, the maximal message length should be $2^{128}$ message blocks of 256 bits each (minus padding) to prevent counter overflow.
Also an attack who intercepts a hash $h$ and knows one message block and its position is easily able to replace it by calculating $h' = h\ \oplus\ ChaCha(m, c)\ \oplus\ ChaCha(m', c)$.
However, if the attack want's to perform a second preimage attack on $h$, he basically would have to find a $m$ with $ChaCha(m, 0) = h$, which is as hard as recovering a key from a 512-bit ChaCha keystream. The attacker can control the counter by adding dummy-blocks, which may give him an advantage given that reduced-round versions of ChaCha have differential characteristics.
However, when it comes to finding any collision, I'm not sure whether you would be able to find a collision with less than $2^{256}$ (birthday attack) calls to the compression functions.