  Adjoining to the language of rings the function symbols for splitting
coefficients, the function symbols for relative $p$-coordinate functions, and
the division predicate for a valuation, some theories of pseudo-algebraically
closed non-trivially valued fields admit quantifier elimination. It is also
shown that in the same language the theory of pseudo-algebraically closed
non-trivially valued fields of a given exponent of imperfection does not admit
quantifier elimination, due to Galois theoretic obstructions.
