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We present JADES JWST/NIRSpec spectroscopy of GN-z11, the most luminous candidate \(z>10\) Lyman break galaxy in the GOODS-North field with \(M_{UV}=-21.5\). We derive a redshift of \(z=10.603\) (lower than previous determinations) based on multiple emission lines in our low and medium resolution spectra over \(0.8-5.3 \mu\)m. We significantly detect the continuum and measure a blue rest-UV spectral slope of \(\beta=-2.4\). Remarkably, we see spatially-extended Lyman-\(\alpha\) in emission (despite the highly-neutral IGM expected at this early epoch), offset 555 km s\(^{-1}\) redward of the systemic redshift. From our measurements of collisionally-excited lines of both low- and high-ionization (including [O II]\(\lambda3727\), [Ne III]\(\lambda 3869\) and C III]\(\lambda1909\)) we infer a high ionization parameter (\(\log U\sim -2\)). We detect the rarely-seen N IV]\(\lambda1486\) and N III]\(\lambda1748\) lines in both our low and medium resolution spectra, with other high ionization lines seen in the low resolution spectrum such as He II (blended with O III]) and C IV (with a possible P-Cygni profile). Based on the observed rest-UV line ratios, we cannot conclusively rule out photoionization from AGN, although the high C III]/He II and N III]/He II ratios are compatible with a star-formation explanation. If the observed emission lines are powered by star formation, then the strong N III]\(\lambda1748\) observed may imply an unusually high \(N/O\) abundance. Balmer emission lines (H\(\gamma\), H\(\delta\)) are also detected, and if powered by star formation rather than an AGN we infer a star formation rate of \(\sim 20-30 M_{\odot} yr^{-1}\) (depending on the IMF) and low dust attenuation. Our NIRSpec spectroscopy confirms that GN-z11 is a remarkable galaxy with extreme properties seen 430 Myr after the Big Bang.