(Abridged) The quiet M2.5 star GJ 436 hosts a warm Neptune that displays an extended atmosphere that dwarfs its own host star. Predictions of atmospheric escape in such planets state that H atoms escape from the upper atmosphere in a collisional regime and that the flow can drag heavier atoms to the upper atmosphere. It is unclear, however, what astrophysical mechanisms drive the process. Our objective is to leverage the extensive coverage of HST/COS observations of the far-ultraviolet (FUV) spectrum of GJ 436 to search for signals of metallic ions in the upper atmosphere of GJ 436 b. We analyzed flux time-series of species present in the FUV spectrum of GJ 436, as well as the Lyman-$alpha$ line. GJ 436 displays FUV flaring events with a rate of $sim$10 d$^{-1}$. There is evidence for a possibly long-lived active region or longitude that modulates the FUV metallic lines of the star with amplitudes up to 20%. Despite the strong geocoronal contamination in the COS spectra, we detected in-transit excess absorption signals of $sim$50% and $sim$30% in the blue and red wings, respectively, of the Lyman-$alpha$ line. We rule out a wide range of excess absorption levels in the metallic lines of the star during the transit. The large atmospheric loss of GJ 436 b observed in Lyman-$alpha$ transmission spectra is stable over the timescale of a few years, and the red wing signal supports the presence of a variable hydrogen absorption source besides the stable exosphere. The previously claimed in-transit absorption in the Si III line is likely an artifact resulting from the stellar magnetic cycle. The non-detection of metallic ions in absorption could indicate that the escape is not hydrodynamic or that the atmospheric mixing is not efficient in dragging metals high enough for sublimation to produce a detectable escape rate of ions to the exosphere.
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