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Invented by Kurt Hensel at the very end of 19th century on the model of powerseries in one indeterminate, the $p$-adic numbers have not only become anindispensable tool of contemporary arithmetic, but a research topic per se. Inthis text, stemming out two talks at the 2023 X-UPS lectures, I shall explaintheir construction, how they fit in a vaster framework, between analysis andarithmetic, where two constructions bearing the name of Isaac Newton play acentral role : Newton's method and Newton's polygon.  Invent\'es par Kurt Hensel \`a la toute fin du 19e si\`ecle sur le mod\`eledes s\'eries formelles en une ind\'etermin\'ee, les nombres $p$-adiques sontdevenus non seulement un outil indispensable de l'arithm\'etique contemporaine,mais un sujet d'\'etude en soi. Dans ce texte, issu de deux expos\'es auxjourn\'ees X-UPS 2023, j'expliquerai leur construction, comment ilss'ins\`erent dans un cadre plus vaste, entre analyse et arithm\'etique, o\`udeux constructions portant le nom d'Isaac Newton jouent un r\^ole central: lam\'ethode de Newton et la notion de polygone de Newton.

Balade newtonienne entre analyse et arithm\'etique (Newtonian promenade between analysis and arithmetic)