  Aims: Imaging and spectroscopy of X-ray extended sources require a proper
characterisation of a spatially unresolved background signal. This background
includes sky and instrumental components, each of which are characterised by
its proper spatial and spectral behaviour. While the X-ray sky background has
been extensively studied in previous work, here we analyse and model the
instrumental background of the ACIS-I detector on-board the Chandra X-ray
observatory in very faint mode.
  Methods: Caused by interaction of highly energetic particles with the
detector, the ACIS-I instrumental background is spectrally characterised by the
superposition of several fluorescence emission lines onto a continuum. To
isolate its flux from any sky component, we fitted an analytical model of the
continuum to observations performed in very faint mode with the detector in the
stowed position shielded from the sky, and gathered over the eight year period
starting in 2001. The remaining emission lines were fitted to blank-sky
observations of the same period. We found 11 emission lines. Analysing the
spatial variation of the amplitude, energy and width of these lines has further
allowed us to infer that three lines of these are presumably due to an energy
correction artefact produced in the frame store.
  Results: We provide an analytical model that predicts the instrumental
background with a precision of 2% in the continuum and 5% in the lines. We use
this model to measure the flux of the unresolved cosmic X-ray background in the
Chandra deep field south. We obtain a flux of $10.2^{+0.5}_{-0.4} \times
10^{13}$ $erg$ $cm^{-2} deg^{-2} s^{-1}$ for the $[1-2]$ keV band and $(3.8 \pm
0.2) \times 10^{12}$ $erg$ $cm^{-2} deg^{-2} s^{-1}$ for the $[2-8]$ keV band.
