= Solution
On the unbounded domain $\mathbb R^3$, translations already prevent <compactness> in every indicated <Hölder space>. Choose a <bump function> $\eta\in C_c^\infty(B_1)$ with $\eta(0)=1$, put $e_1=(1,0,0)$, and define
$$
\boxed{u_m(x)=\eta(x-3m e_1),\qquad m=1,2,\ldots.}
$$
A <change of variables> in the <Lebesgue integrals> defining the <Sobolev norm> gives $\|u_m\|_{W^{1,6}(\mathbb R^3)}=\|\eta\|_{W^{1,6}(\mathbb R^3)}$, so the sequence is bounded. Each $u_m$ is a <smooth function> of <compact support> and belongs to every <Hölder space> $C^{0,\gamma}(\mathbb R^3)$, $0<\gamma\leq1$: its bounded <gradient> gives the <Lipschitz> estimate for distances at most one, and its bounded values control larger distances.
For $m\ne\ell$, the <supports> are disjoint, and at $x=3m e_1$ we have $u_m(x)=1$ and $u_\ell(x)=0$. Therefore
$$
\|u_m-u_\ell\|_{C^{0,\gamma}}\geq\|u_m-u_\ell\|_\infty\geq1
\qquad(0<\gamma\leq1).
$$
No <subsequence> can be a <Cauchy sequence> in any of these <Hölder norms>, hence none converges. This uses the translation mechanism for <noncompactness of a Sobolev embedding by translation>; concentration at the critical <Hölder exponent> alone would not rule out convergence at every smaller <Hölder exponent>.
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