  We decompose the restriction of ramified principal series representations of
the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup
$K=\mathrm{GL}(3,\mathscr{R})$. Its decomposition is dependent on the degree of
ramification of the inducing characters and can be characterized in terms of
filtrations of the Iwahori subgroup in $K$. We establish several irreducibility
results and illustrate the decomposition with some examples.
