  The detailed composition of most metal-poor halo stars has been found to be
very uniform. However, a fraction of 20-70% (increasing with decreasing
metallicity) exhibit dramatic enhancements in their abundances of carbon - the
so-called carbon-enhanced metal-poor (CEMP) stars. A key question for Galactic
chemical evolution models is whether this non-standard composition reflects
that of the stellar natal clouds, or is due to local, post-birth mass transfer
of chemically processed material from a binary companion; CEMP stars should
then all be members of binary systems. Our aim is to determine the frequency
and orbital parameters of binaries among CEMP stars with and without
over-abundances of neutron-capture elements - CEMP-s and CEMP-no stars,
respectively - as a test of this local mass-transfer scenario. This paper
discusses a sample of 24 CEMP-no stars, while a subsequent paper will consider
a similar sample of CEMP-s stars. Most programme stars exhibit no statistically
significant radial-velocit variation over this period and appear to be single,
while four are found to be binaries with orbital periods of 300-2,000 days and
normal eccentricity; the binary frequency for the sample is 17+-9%. The single
stars mostly belong to the recently-identified ``low-C band'', while the
binaries have higher absolute carbon abundances. We conclude that the
nucleosynthetic process responsible for the strong carbon excess in these
ancient stars is unrelated to their binary status; the carbon was imprinted on
their natal molecular clouds in the early Galactic ISM by an even earlier,
external source, strongly indicating that the CEMP-no stars are likely bona
fide second-generation stars. We discuss potential production sites for carbon
and its transfer across interstellar distances in the early ISM, and
implications for the composition of high-redshift DLA systems. Abridged.
