This is not an answer to your question, but I found it a few days ago looking for something else and did strike me as quite curious... :

According to E. E. Enochs [A note on reflexive modules. Pacific J. Math. Volume 14, Number 3 (1964), 879-881.] a free module of infinite countable rank over a discrete valuation ring is reflexive iff the ring is not complete.

A nice related example to keep in mind in this context is that a free abelian group of infinite countable rank is reflexive.

This is not an answer to your question, but I found it a few days ago looking for something else and did strike me as quite curious... :

According to E. E. Enochs [A note on reflexive modules. Pacific J. Math. Volume 14, Number 3 (1964), 879-881.] a free module of infinite countable rank over a discrete valuation ring is reflexive iff the ring is not complete.

A nice related example to keep in mind in this context is that a free abelian group of infinite countable rank is reflexive.