  We prove the arithmetic fundamental lemma conjecture over a general $p$-adic
field with odd residue cardinality $q\geq \dim V$. Our strategy is similar to
the one used by the second author during his proof of the AFL over
$\mathbb{Q}_p$ (arXiv:1909.02697), but only requires the modularity of divisor
generating series on the Shimura variety (as opposed to its integral model).
The resulting increase in flexibility allows us to work over an arbitrary base
field. To carry out the strategy, we also generalize results of Howard
(arXiv:1303.0545) on CM-cycle intersection and of Ehlen--Sankaran
(arXiv:1607.06545) on Green function comparison from $\mathbb{Q}$ to general
totally real base fields.
