The paper studies the topological cyclic homology functor of rings. This associates to a ring $R$ a spectrum $TC(R)$ which turns out to be closely related to Quillen's $K(R)$, but which is better suited for algebraic topological analysis. The homotopy groups of $TC(R)$ may be viewed as a topological refinement of Connes' cyclic homology groups. For rings of integers in local number fields with residue characteristic $p > 0$, a recent result of R. McCarthy implies that the cyclotomic trace from $K(R)$ to $TC(R)$ becomes a homotopy equivalence after completion at $p$. In particular this is so when $R$ is the ring of $p$-adic integers. Our principal result evaluates the $p$-adic homotopy type of $TC({\mathbb Z}_p)$ when $p$ is odd, modulo a certain conjecture, and we give evidence to support the conjecture. It appears that S. Tsalidis has now settled the conjecture, his arguments in part being based upon the analysis presented in this paper. The $p$-completion of $TC({\mathbb Z}_p)$ turns out to be the product of three spectra, namely the $p$-completion of the special unitary group $SU$, the $p$-completion of Quillen's $F\psi k$ , also called the image of $J$-space, and the $p$-completion of it ifying space. Here $k$ is a generator of the $p$-adic units. This then determines the homotopy groups with $p$-adic coefficients of $K({\mathbb Z}_p)$. The methods of the paper are homotopy theoretical, and even involve equivariant homotopy theory with respect to certain natural circle actions which go back to Connes' theory of cyclic sets. The main tools are spectral sequences and other methods from algebraic topology.