These answers provides two examples of non barrelled locally convex spaces, but they are both incomplete. Furthermore, their completion is a Banach space and so necessarily barrelled.
This answer also presents an incomplete example (which is not normed), but its completion is also barrelled.
Is there any example of a complete, non barrelled, locally convex space?
clarification: By complete locally convex space I mean a space in which every Cauchy net converges (like in Kelley's General Topology p. 192).