This paper is motivated by the non-Archimedean counterpart of a problem raised by Mahler and Mendès France, and by questions related to the expected normality of irrational algebraic numbers. We introduce a class of sequence enjoying a particular combinatorial property: the precocious occurrences of infinitely many symmetric patterns. Then, we prove some transcendence statements involving both real and $p$-adic numbers associated with such palindromic sequences.