We study the asymptotic behaviour of the perturbative series in the heavy quark effective theory (HQET) using the 1/Nƒ expansion. We find that this theory suffers from an ultraviolet renormalon problem, corresponding to a non-Borel-summable behaviour of perturbation series in large orders, and leading to a principal nonperturbative ambiguity in its definition. This ambiguity is related to an infrared renormalon in the pole mass and can be understood as the necessity to include the residual mass term δm in the definition of HQET, which must be considered as ambiguous (and possibly complex), and is required to cancel the ultraviolet renormalon singularity generated by the perturbative expansion. The formal status of δm is thus identical to that of condensates in the conventional short-distance expansion of correlation functions in QCD. The status of the pole mass of a heavy quark, the operator product expansion for inclusive decays, and QCD sum rules in the HQET are discussed in this context.