secp256k1 prime modulus vs order
Yes, field operations may involve numbers that are larger than the order. Public key point coordinates may exceed the order.
Parts of the protocol that handle scalars work modulo the curve order, while the parts that handle elliptic curve points work modulo the prime modulus.
Sometimes, as you noted, some information &quot;crosses&quot; between these domains: the main case being inside the ECDSA protocol, in the r value which is obtained from a X coordinate and reduced modulo the order to enter in the computation of s.