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We decompose the restriction of ramified principal series representations of the 
$p$-adic group 
$\text{GL}\left( 3,\,\text{k} \right)$ to its maximal compact subgroup 
$K\,=\,\text{GL}\left( 3,\,\mathcal{R} \right)$. 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.
