There are lots of criticisms of p-values, and while I don’t agree with all of them, it is important to understand these criticisms. This article discusses the definition of p-values commonly used by many people (including me) as “measures of evidence.” If you define the concept of “measures of evidence” carefully, the p-value fails to meet this definition.
Daniel Lakens. Why p-values are not measures of evidence, Trends in Ecology and Evolution. Available in pdf format.