According to standard lore, perturbative series of super-renormalizable theories have only instanton singularities. In this paper we show that two-dimensional scalar theories with a spontaneously broken O(N) symmetry at the classical level, which are super-renormalizable, have an IR renormalon singularity at large N. Since perturbative expansions in these theories are made around the "false vacuum" in which the global symmetry is broken, this singularity can be regarded as a manifestation of the non-perturbative absence of Goldstone bosons. We conjecture that the Borel singularity in the ground state energy of the Lieb--Liniger model is a non-relativistic manifestation of this phenomenon. We also provide {\it en passant} a detailed perturbative calculation of the Lieb--Liniger energy up to two-loops, and we check that it agrees with the prediction of the Bethe ansatz.
