  Using topological cyclic homology, we give a refinement of Beilinson's
$p$-adic Goodwillie isomorphism between relative continuous $K$-theory and
cyclic homology. As a result, we generalize results of Bloch-Esnault-Kerz and
Beilinson on the $p$-adic deformations of $K$-theory classes. Furthermore, we
prove structural results for the Bhatt-Morrow-Scholze filtration on $TC$ and
identify the graded pieces with the syntomic cohomology of Fontaine-Messing.
