  Using $L^2$-methods for the $\bar\partial$-equation we prove that the
Ohsawa-Takegoshi extension theorem also holds for holomorphic sections of a
vector bundle, over compact K\"ahler manifolds. We then proceed to show that
the conditions that are needed are more liberal than the ones one would need if
one instead reduced the extension problem to line bundles through the usual
algebraic geometric procedure of studying the projective bundle associated with
the vector bundle.
