<p>In Kaluza-Klein theory, gauge fields on <i>M</i>₄ arise as components of a higher-dimensional metric defined on <i>M</i>₄ × <i>K</i>. The traditional expectation is that all the gauge fields of the Standard Model are linked to exact Killing vector fields on the internal space. This paper questions that assumption and investigates the properties of 4D gauge fields linked to non-Killing fields on <i>K</i>. It is shown that they have massive yet arbitrarily light bosons; they can mix fermions with different masses; and they can have asymmetric couplings to left- and right-handed fermions. None of these properties is easily satisfied by gauge fields linked to internal isometries. Thus, backgrounds encoding such massive gauge fields circumvent traditional no-go arguments and offer a geometric source of chiral interactions with fermions. This may help to model the weak force within the Kaluza-Klein framework. Technically, the paper uses the language of spin geometry and Riemannian submersions. It studies the higher-dimensional Dirac operator with non-trivial background metrics. The results are derived for a general <i>K</i>. They are illustrated explicitly in the simpler cases where <i>K</i> is the two-sphere and the two-torus.</p>