possibility of metallic ammonium
A nomenclature note: conventionally, in chemistry "a metal" is an element (such as a transition metal or alkali metal). If what you're talking about is a substance that behaves as a metal, that's typically referred to as metallic, like metallic glasses. 

Therefore, I would rephrase your question as "Can metallic ammonium exist?" - in fact, this has been a question for quite awhile. From what I've read, there's a theoretical basis that it should be possible at given temperatures and pressures, but it hasn't been experimentally observed. The papers I was reading were from the 50s-70s, so I'll keep looking for more recent information. 

Solvated electrons are slightly related, in that there is a "free" electron, and this was in fact originally documented in liquid ammonia.