Does every non-Archimedean absolute value satisfy the ultrametric inequality?
Yes, a non-Archimedean absolute value must be ultrametric. See e.g. Theorem 2.1 of these notes.
Indeed, a non-Archimedean absolute value automatically satisfies the ultrametric inequality (as pointed out by Robert Israel). In my original question, I used a slightly different formulation of the Archimedean property (and the referenced lecture notes might not be online forever), so here is a full proof.
Proposition. Let $F$ be a field and let $|\cdot|$ be a non-Archimedean absolute value. Then $|\cdot|$ satisfies the ultrametric inequality.
Proof. Since $|\cdot|$ is non-Archimedean, we may choose some non-zero $x\in F$ such that $$ \big|\:\underbrace{x + \cdots + x}_{n\ \text{times}}\:\big| \leq 1,\tag*{for all $n\in\mathbb{N}$.} $$ We may interpret any element of $\mathbb{N}$ (or $\mathbb{Z}$, for that matter) as an element of $F$ by identifying it with its image under the natural ring homomorphism $\mathbb{Z} \to F$. Then the above becomes $$ |n|\cdot |x| = |n\cdot x| \leq 1,\tag*{for all $n\in\mathbb{N}$.} $$ Since $x$ is non-zero by assumption, we have $|x| \neq 0$, hence $$ |n| \leq \frac{1}{|x|},\tag*{for all $n\in\mathbb{N}$.} $$ Now let $y,z\in F$ be given. By the binomial theorem, for all $k\in\mathbb{N}$ we have $$ (y + z)^k \: = \: \sum_{j=0}^k \binom{k}{j} y^j z^{k-j}, $$ hence $$ |y+z|^k \: = \: |(y+z)^k| \: = \: \left|\sum_{j=0}^k \binom{k}{j} y^j z^{k-j}\right| \: \leq \: \sum_{j=0}^k \frac{|y|^j\cdot |z|^{k-j}}{|x|} \: \leq \: \frac{k+1}{|x|}\cdot \max(|y|,|z|)^k. $$ Equivalently: for all $k\in\mathbb{Z}_{> 0}$ we have $$ |y+z| \: \leq \: \sqrt[k]{\frac{k+1}{|x|}}\cdot \max(|y|,|z|). $$ As $k$ increases, this factor $\sqrt[k]{\frac{k+1}{|x|}}$ converges (decreasingly) to one, so we have $$ |y + z| \: \leq \: \inf_{k\to\infty} \sqrt[k]{\frac{k+1}{|x|}}\cdot \max(|y|,|z|) \: = \: \max(|y|,|z|).\tag*{$\Box$} $$
This peculiar little trick is now standard in the literature. It is also used in many textbooks, for instance:
Paulo Ribenboim, The Theory of Classical Valuations, Springer Monographs in Mathematics (section 1.2, fact E).
Antonio J. Engler & Alexander Prestel, Valued Fields, Springer Monographs in Mathematics (proposition 1.1.1).
Pierre Antoine Grillet, Abstract Algebra, Second Edition, Springer Graduate Texts in Mathematics 242 (chapter VI, proposition 3.2).
Alain M. Robert, A Course in p-adic Analysis, Springer Graduate Texts in Mathematics 198 (chapter 2, section 1.6, first theorem).