Unit balls with precisely the property that you are looking for have been studied under the rather awkward name of completant (presumably directly from the French) in the book on applications of bornologies to functional analysis by Hogbe-Nlend. I think that the only result of any substance that you will find is  a variant of Grothendieck's completeness theorem which can be found there.  One assumes that the ball is a closed bounded set in an ambient topological vector space which is complete.  This, amongst others, provides what is probably the simplest and most transparent proof of the completeness of the $  \ell^p$ and $L^p  $-spaces.

By the way the class of spaces of Bill Johnson's answer has also been investigated.  They were introduced by Waelbroeck and called Waelbroeck spaces by Buchwalter.  They form a concrete representation of the category opposite to that of Banach spaces---see Cigler-Losert-Michor on functors on categories of Banach spaces (available onine).  A good example of their use is in the characterisation of von Neumann algebas as $ C^*$ algebras which, as Banach spaces, are Waelbroeck.  This givea a useful pointer on how to form limits in the category of von Neumann algebras.