In Jarchow's Locally Convex Spaces this not being quasi-complete is asserted on page 206 referring to Corollary 11.4.4 on page 228 saying that a Banach space is reflexive if and only if its closed unit ball is weakly (sequentially/countably) compact.
I see how the compactness property implies the quasi-completeness, but I do not see how the converse (that is needed here) would be deduced.
Is the asserted non-quasi-completeness "well-known", and is there a reference containing a proof, or is Jarchow's assertion false?
Added. (25.6.2017) Having read certain places in R. E. Edwards' Functional Analysis, J. Horváth's Topological Vector Spaces and H. Jarchow's Locally Convex Spaces, I have now found (from my head) the following result:
If someone can give a precise reference to an explicitly formulated previously published result with proper proof (not any vague explanations) of a result with the same content as Proposition 4 above, I am still interested to know. If it is posted as an answer, and I can easily check it via the Internet, I will accept it.