C-minimality is a model-theoretical notion, introduced by Haskell, Macpherson and Steinhorn around 1994. It is an adaption of strong minimality to ultrametric spaces, as o-minimality was an adaptation to linear orders. Algebraically closed valued fields are the typical example. We generalize slightly the original definition by allowing isolated points in the structure. This makes the comparison easier to o-minimal structures. For example a C-minimal structure is definably decomposable into a dense part and a discrete one, exactly as an o-minimal structure is. But the two parts must not be orthogonal, as they must be in an o-minimal structure. We also consider the question of the differentiability of functions definable in a C-minimal field.