  We prove that for p-optimal fields (a very large subclass of p-minimal fields
containing all the known examples) a cell decomposition theorem follows from
methods going back to Denef's paper [Invent. Math, 77 (1984)]. We derive from
it the existence of definable Skolem functions and strong p-minimality. Then we
turn to strongly p-optimal fields satisfying the Extreme Value Property (a
property which in particular holds in fields which are elementarily equivalent
to a p-adic one). For such fields K, we prove that every definable subset of
KxK^d whose fibers are inverse images by the valuation of subsets of the value
group, are semi-algebraic. Combining the two we get a preparation theorem for
definable functions on p-optimal fields satisfying the Extreme Value Property,
from which it follows that infinite sets definable over such fields are
isomorphic iff they have the same dimension.
