An inclusive search for anomalous production of two prompt, isolated muons with the same electric charge is presented. The search is performed in a data sample corresponding to $1.6\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity collected in 2011 at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$ with the ATLAS detector at the LHC. Muon pairs are selected by requiring two isolated muons of the same electric charge with ${p}_{\mathrm{T}}>20\text{ }\text{ }\mathrm{GeV}$ and $|\ensuremath{\eta}|<2.5$. Minimal requirements are placed on the rest of the event activity. The distribution of the invariant mass of the muon pair $m(\ensuremath{\mu}\ensuremath{\mu})$ is found to agree well with the background expectation. Upper limits on the cross section for anomalous production of two muons with the same electric charge are placed as a function of $m(\ensuremath{\mu}\ensuremath{\mu})$ within a fiducial region defined by the event selection. The fiducial cross-section limit constrains the like-sign top-quark pair-production cross section to be below 3.7 pb at 95% confidence level. The data are also analyzed to search for a narrow like-sign dimuon resonance as predicted for e.g. doubly charged Higgs bosons (${H}^{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}$). Assuming pair production of ${H}^{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}$ bosons and a branching ratio to muons of 100% (33%), this analysis excludes masses below 355 (244) GeV and 251 (209) GeV for ${H}^{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}$ bosons coupling to left-handed and right-handed fermions, respectively.