Context. The space based γ-ray observatory INTEGRAL of the European Space Agency (ESA) includes the spectrometer instrument “SPI”. This is a coded mask telescope featuring a 19-element Germanium detector array for high-resolution γ-ray spectroscopy, encapsulated in a scintillation detector assembly that provides a veto for background from charged particles. In space, cosmic rays irradiate spacecraft and instruments, which, in spite of the vetoing detectors, results in a large instrumental background from activation of those materials, and leads to deterioration of the charge collection properties of the Ge detectors.Aim. We aim to determine the measurement characteristics of our detectors and their evolution with time, that is, their spectral response and instrumental background. These incur systematic variations in the SPI signal from celestial photons, hence their determination from a broad empirical database enables a reduction of underlying systematics in data analysis. For this, we explore compromises balancing temporal and spectral resolution within statistical limitations. Our goal is to enable modelling of background applicable to spectroscopic studies of the sky, accounting separately for changes of the spectral response and of instrumental background.Methods. We use 13.5 years of INTEGRAL/SPI data, which consist of spectra for each detector and for each pointing of the satellite. Spectral fits to each such spectrum, with independent but coherent treatment of continuum and line backgrounds, provides us with details about separated background components. From the strongest background lines, we first determine how the spectral response changes with time. Applying symmetry and long-term stability tests, we eliminate degeneracies and reduce statistical fluctuations of background parameters, with the aim of providing a self-consistent description of the spectral response for each individual detector. Accounting for this, we then determine how the instrumental background components change in intensities and other characteristics, most-importantly their relative distribution among detectors.Results. Spectral resolution of Ge detectors in space degrades with time, up to 15% within half a year, consistently for all detectors, and across the SPI energy range. Semi-annual annealing operations recover these losses, yet there is a small long-term degradation. The intensity of instrumental background varies anti-correlated to solar activity, in general. There are significant differences among different lines and with respect to continuum. Background lines are found to have a characteristic, well-defined and long-term consistent intensity ratio among detectors. We use this to categorise lines in groups of similar behaviour. The dataset of spectral-response and background parameters as fitted across the INTEGRAL mission allows studies of SPI spectral response and background behaviour in a broad perspective, and efficiently supports precision modelling of instrumental background.