Is there a nonprojective flat module over a local ring? Here I assume the ring is commutative with unit.
$\mathbb{Q}$ is flat over $\mathbb{Z}_p$, but not projective. 


It is related to Bass' theorem. Flat modules are projective iff the ring is perfect. $p$adic integers or formal power series are examples of local rings which are not perfect and have nonprojective flat modules. 

