They already exist, there's a small fruit-named company that happens to make them.

Are you thinking of http://en.wikipedia.org/wiki/Bada ? I've never heard of a purpose-built BSD phone other than those (although there are plenty of unofficial ports)

iOS is based on OS X/Darwin which is in turn based on BSD.

Darwin itself uses very little BSD code. It's built on Mach. OS X does however import much of the BSD userland, but I don't even know if much of that makes it into iOS.

I think it'd be a fair guess to say that iOS likely contains very little BSD Unix code.

For what it's worth, Android also uses a lot of BSD userspace code (see toolbox).