 Solution

For any injection $u:U\hookrightarrow V$, the direct sum decomposition $F=P\oplus Q$ gives natural module isomorphismsUnder these, $u\otimes 1_F$ is the block-diagonal map $(u\otimes 1_P)\oplus (u\otimes 1_Q)$. It is injective because $F$ is a flat module. If an element lies in the kernel of $u\otimes 1_P$, insert zero in the other component; injectivity of the whole map forces that element to be zero. Hence $u\otimes 1_P$ is injective for every $u$, which proves $\boxed{P\text{ is flat}}$. This establishes that direct summands of flat modules are flat, without a finite-generation assumption.