  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.
