If you are truly only interested in the triangle inequality, then there is the Hamming weight
$w(x_1,x_2,\ldots,x_n)=m$ where $m$ is simply the number of non-zero components. This gives you a metric. Mind you, the space $F^n$ is finite, so any metric on it is going to give you the discrete topology. Adding any kind of norm-like requirements (on top of the triangle inequality) is problematic for several reason, as others have pointed out.
The Hamming weight obviously depends on the choice of basis, which may restrict its usefulness (depending on what you wanted to do with this 'norm').
I'm sad to say there isn't an awful lot of analysis happening in this space.