This Letter presents the first study of $W^{\pm}W^{\pm}jj$, same-electric-charge diboson production in association with two jets, using 20.3 fb$^{-1}$ of proton--proton collision data at $\sqrt{s}=8$ TeV recorded by the ATLAS detector at the Large Hadron Collider. Events with two reconstructed same-charge leptons ($e^\pm e^\pm$, $e^\pm \mu^\pm$, and $\mu^\pm \mu^\pm$) and two or more jets are analyzed. Production cross sections are measured in two fiducial regions, with different sensitivities to the electroweak and strong production mechanisms. First evidence for $W^{\pm}W^{\pm}jj$ production and electroweak-only $W^{\pm}W^{\pm}jj$ production is observed with a significance of $4.5$ and $3.6$ standard deviations respectively. The measured production cross sections are in agreement with Standard Model predictions. Limits at 95% confidence level are set on anomalous quartic gauge
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