Given that an attacker then knows the values of $m$, $c$, $e$ and $n$ in the following equation, is there any extensions which allows them to recover $d$?
$c = m^e \mod n$
No, an attack against RSA without padding that let one recover messages does not, as far as we know, degenerate into leaking a working private exponent $d$, or factoring $n$. In fact, even unlimited access to a decryption oracle performing $c\mapsto m=c^d\bmod n$ for any $c$ will not allow that either.