During Winter 2020 I teach a course on Berkovich spaces (Math 715). Here is the syllabus.Currently I am working on a book on Berkovich spaces, including some of the background on semi-normed commutative algebra and, specifically, affinoid algebras. Portions of this book will be distributed during the course (see below). Otherwise, the three general reference for the course is the book and IHES paper by Berkovich himself, and the book known as BGR.
Here are some annotations where I fill in various details and make additional comments on the first two chapters in Berkovich's book.
These notes by Brian Conrad give a very nice introduction to non-Archimedean geometry in general. Similarly, these notes by Michael Temkin are useful for learning about Berkovich spaces.
These notes that I wrote for a summer school in 2010 have dynamics as a focus but contain a reasonably detailed discussion of tree structures and potential theory on trees. This is relevant for the Berkovich disc and projective line.
Parts of book (under construction; comments are welcome!):