  We establish H\"ormander-type $L^2$-estimates for the
$\overline{\partial}$-operators that hold uniformly for all nontrivial flat
holomorphic line bundles on compact K\"ahler manifolds. Our result can be
regarded as a $\overline{\partial}$-version of Ueda's lemma on the operator
norm of \v{C}ech coboundaries for flat line bundles and indeed recovers the
original version of Ueda's lemma for compact K\"ahler manifolds. A partial
generalisation for $(p,0)$-forms on Ricci-flat manifolds is also given.
