In RSA, is it possible to compute the private key if one knows one of the primes used to generate n? Within RSA the private key can be derived if one knows both $p$ and $q$. Is it possible to derive the RSA private key or decrypt / learn something about encrypted messages if you only know one of the prime numbers and not the modulus $n=pq$? The only available knowledge of the crypto-system is the ciphertext of a single encrypted message and the exponent ($e$) used.

EDIT: In actuality what I want to do is to retrieve the original message $m$ that was encrypted with the private key resulting from $p$ and $q$. I assumed that finding the original message would be equal in difficulty to calculating the private key.
I'll read the problem as stating we have $c$, $e$ and $p$ from some encryption $c=m^e\bmod(pq)$. But for some reason we do not have $n$ (even thought it is part of the public key, which is assumed public).

We know the order of magnitude of $q$, which likely is little above $\log_2(c)-\log_2(p)$-bit. But that's a far cry from $q$.

We can compute $d_p=e^{-1}\bmod p$, then $m_p=c^{d_p}\bmod p$ which is $m\bmod p$. Depending on context, that might let us guess $m$. For textbook RSA encryption and a narrow message (bit size of $m$ less or comparable to the bitsize of $p$), we will have $m=m_p$ or $m=m_p+k\,p$ for some small $k$, revealing $m$ or allowing to guess $k$ and $m$ if $m$ has some redundancy.

If we somewhat get the full $m$, we can compute $a=(m^e-c)/p$ and that's a multiple of $q$. If we are lucky, $a$ can be fully factored, revealing $q$. That's not entirely impossible for very small $e$ when using ECM to try to pull the factors of $a$.

If we can similarly find the full $m'$ for another $c'$, then we can compute $a'=(m'^e-c')/p$ and then compute $\gcd(a,a')$. That's a multiple of $q$, and has an excellent chance to be easy to fully factor, revealing $q$.

With $p$ and $q$, we can compute a working private key per any of the usual methods in RSA.