This Letter presents a search for a hidden-beauty counterpart of the $X(3872)$ in the mass ranges 10.05--10.31 GeV and 10.40--11.00 GeV, in the channel $X_b \to \pi^{+} \pi^{-} \Upsilon(1 \rm S)(\rightarrow\mu^+\mu^-)$, using 16.2 fb$^{-1}$ of $\sqrt{s}=8$ TeV $pp$ collision data collected by the ATLAS detector at the LHC. No evidence for new narrow states is found, and upper limits are set on the product of the $X_b$ cross section and branching fraction, relative to those of the $\Upsilon(2 \rm S)$, at the 95% confidence level using the CL$_S$ approach. These limits range from 0.8% to 4.0%, depending on mass. For masses above 10.1 GeV, the expected upper limits from this analysis are the most restrictive to date. Searches for production of the $\Upsilon(1^3 \rm{D}$$_J)$, $\Upsilon(10860)$, and $\Upsilon(11020)$ states also reveal no significant