  For any unramified quadratic extension of $p$-adic local fields $F/F_0$
$(p>2)$, we formulate several arithmetic transfer conjectures at any maximal
parahoric level, in the context of Zhang's relative trace formula approach to
the arithmetic Gan--Gross--Prasad conjecture. The formulation involves a way to
resolve the singularity of relevant moduli spaces via natural stratifications
and modify derived fixed points. By a local-global method and double induction,
we prove these conjectures for $F_0$ unramified over $\mathbb Q_p$, including
the arithmetic fundamental lemma for $p>2$. Moreover, we prove new modularity
results for arithmetic theta series at parahoric levels via a method of
modification over $\mathbb F_q$ and $\mathbb C$. Along the way, we study the
complex and mod $p$ geometry of Shimura varieties and special cycles. We
introduce the relative Cayley map and also establish Jacquet--Rallis transfers
at maximal parahoric levels.
