Summary: Let \(p\) be a prime number, \(\mathbb Q_p\) the field of \(p\)-adic numbers, \(\overline{\mathbb Q}_p\) a fixed algebraic closure of \(\mathbb Q_p\) and \(\mathbb C_p\) the completion of \(\overline{\mathbb Q}_p\) with respect to the \(p\)-adic valuation. Let \(G_p = Gal_{cont} (\mathbb C_p/\mathbb Q_p)\) be the group of continuous automorphisms of \(\mathbb C_p\) over \(\mathbb Q_p\). We investigate isometric Galois actions of the Galois group \(G_p\) on subsets of \(\mathbb C_p\).