A robust sign test is proposed for testing unit roots in cross-sectionally dependent panel data. Large sample Gaussian null asymptotics of the test are established under (fixed N, large T) and, for serially uncorrelated error cases, under (large N, fixed T), where N is the number of panel units and T is the length of time span. The limiting null distribution is valid, even if the error processes are subject to any type of conditional heteroscedasticity. A Monte-Carlo experiment reveals that, compared with other existing tests, the proposed test has a very stable size property for wider classes of error distributions, type of conditional heteroscedasticities, type of cross-sectional correlations, and values of (N, T) while having reasonable power. Especially, for small T like T = 5, 10, 20, the proposed test shows much stabler size performance than other existing tests. The unemployment rates of the 51 states of the USA are analyzed by the proposed method, which reveals some evidence for unit roots in the presence of factor and spatial cross-section correlation.