Let
F
F
be a complete topological field. We undertake a study of the ring
C
(
X
,
F
)
C(X,F)
of all continuous
F
F
-valued functions on a topological space
X
X
whose topology is determined by
C
(
X
,
F
)
C(X,F)
, in that it is the weakest making each function in
C
(
X
,
F
)
C(X,F)
continuous, and of the ring
C
∗
(
X
,
F
)
{C^\ast }(X,F)
of all continuous
F
F
-valued functions with relatively compact range, where the topology of
X
X
is similarly determined by
C
∗
(
X
,
F
)
{C^\ast }(X,F)
. The theory of uniform structures permits a rapid construction of the appropriate generalizations of the Hewitt realcompactification of
X
X
in the former case and of the Stone-Čech compactification of
X
X
in the latter. Most attention is given to the case where
F
F
and
X
X
are ultraregular; in this case we determine conditions on
F
F
that permit a development parallel to the classical theory where
F
F
is the real number field. One example of such conditions is that the cardinality of
F
F
be nonmeasurable and that the topology of
F
F
be given by an ultrametric or a valuation. Measure-theoretic interpretations are given, and a nonarchimedean analogue of Nachbin and Shirota’s theorem concerning the bornologicity of
C
(
X
)
C(X)
is obtained.