I will explain what's going on. We call $|x|_p$ the $p$-adic absolute value of $x$ and $v_p(x)$ the $p$-adic valuation of $x$. The distinction that is made by the two terms "absolute value" and "valuation" is completely standard... in English. However, Khrennikov is originally from Russia and in Russian there is one term for both concepts (нормирование = normirovanie, with stress -- for English speakers I am not making this up -- on the second syllable). There is a term "absolute value" in Russian, but it is not an abstract concept; it refers only to the usual absolute value on the real or complex numbers (and quaternions?). This is perhaps why Khrennikov is using the term "valuation" incorrectly to refer to an absolute value function.
(I'm giving a course in Moscow this semester and I found this point very confusing frustrating when I was preparing my initial lectures. In different books I wanted to use a concise term found the same word used for an absolute value which would not suggest and for a valuation , but I couldn't figure out how to do it and eventually couldn't find the term that exclusively means absolute value. Eventually I determined it there isn't doneone; you just know by context what meaning is intended. Native speakers are welcome to correct me here.)
