We show, on typical examples, how local objects (i.e. germs of analytic vector fields or diffeomorphisms of $\mathbb {C}^\nu $) which, due to resonance or small denominators, fail to possess an analytic linearisation, may still be reduced to their linear part by means of ramified changes of coordinates. The latter are not merely formal, but canonically resummable in spiral-like neighbourhoods of the ramified origin $0$ of $\mathbb {C}^\nu $. Apart from its obvious bearing on local dynamics, ramified linearisation leads to an extension of the concept of holonomy