As long as the form is positive definite and the unit ball is convex, you get a perfectly good Banach space using any symmetric $n$-linear form on a real vector space $V$.  The degree $n$ is necessarily even.  It is equivalent to defining the norm as the $n$th root of a homogeneous degree $n$ polynomial.  $\ell^p$ is an example for any even integer $p$.   There are many other examples.  I found a paper, [Banach spaces with polynomial norms][1], by Bruce Reznick, that studies these norms.  He obtains various results; the most appealing one to me at a glance is that these Banach spaces are all reflexive.

Off-hand I can't think of any simple way to recover positive definiteness starting with odd polynomials.  The cube of the norm on $\ell^3$ is a polynomial in the absolute values of the coordinates rather than the coordinates themselves.

  [1]: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1102785074&page=record