  We propose a sequential, anytime-valid method to test the conditional
independence of a response $Y$ and a predictor $X$ given a random vector $Z$.
The proposed test is based on e-statistics and test martingales, which
generalize likelihood ratios and allow valid inference at arbitrary stopping
times. In accordance with the recently introduced model-X setting, our test
depends on the availability of the conditional distribution of $X$ given $Z$,
or at least a sufficiently sharp approximation thereof. Within this setting, we
derive a general method for constructing e-statistics for testing conditional
independence, show that it leads to growth-rate optimal e-statistics for simple
alternatives, and prove that our method yields tests with asymptotic power one
in the special case of a logistic regression model. A simulation study is done
to demonstrate that the approach is competitive in terms of power when compared
to established sequential and nonsequential testing methods, and robust with
respect to violations of the model-X assumption.
