  Colliot-Th{\'e}l{\`e}ne has determined the Chow group of zero-cycles on a
Ch{\^a}telet surface X defined over a finite extension K of the field of p-adic
numbers (p an odd prime) when X is split by an unramified extension of K. Using
similar techniques, we extend this calculation to the ramified case,
simplifying his result in the process. We also determine how this group changes
when we pass from K to a finite extension of K.
