  We prove that in every ring of generalised power series with non-positive
real exponents and coefficients in a field of characteristic zero, every series
admits a factorisation into finitely many irreducibles of infinite support, the
number of which can be bounded in terms of the order type of the series, and a
unique product, up to multiplication by a unit, of factors of finite support.
  We deduce analogous results for the ring of omnific integers within Conway's
surreal numbers, using a suitable notion of infinite product. In turn, we solve
Gonshor's conjecture that the omnific integer $\omega^{\sqrt{2}} + \omega + 1$
is prime.
  We also exhibit new classes of irreducible and prime generalised power series
and omnific integers, generalising previous work of Berarducci and Pitteloud.
