This is assuming proper methodology and statistics so that the p-value actually matches the chance of the result arising by chance. In practice, since even your best judgment of the methodology is not going to account for certainty in the soundness of the experiment, I would say that a p-value of 0.001 constitutes considerably less than 10 bits of evidence, because the odds that something was wrong with the experiment are better than the odds that the results were coincidental. Multiple experiments with lower cumulative p-value can still be stronger evidence if they all make adjustments to account for possible sources of error.
This is assuming proper methodology and statistics so that the p-value actually matches the chance of the result arising by chance. In practice, since even your best judgment of the methodology is not going to account for certainty in the soundness of the experiment, I would say that a p-value of 0.001 constitutes considerably less than 10 bits of evidence, because the odds that something was wrong with the experiment are better than the odds that the results were coincidental. Multiple experiments with lower cumulative p-value can still be stronger evidence if they all make adjustments to account for possible sources of error.