We present constraints on extensions to the standard cosmological model of a spatially flat Universe governed by general relativity, a
cosmological constant (Λ), and cold dark matter (CDM) by varying the spatial curvature ΩK, the sum of the neutrino masses Pmν, the dark energy equation of state parameter w, and the Hu-Sawicki f(R) gravity fR0 parameter. With the combined 3 × 2pt measurements of cosmic shear from the Kilo-Degree Survey (KiDS-1000), galaxy clustering from the Baryon Oscillation Spectroscopic Survey
(BOSS), and galaxy-galaxy lensing from the overlap between KiDS-1000, BOSS, and the spectroscopic 2-degree Field Lensing Survey (2dFLenS), we find results that are fully consistent with a flat ΛCDM model with ΩK = 0.011+0.054−0.057,
Pmν < 1.76 eV (95%CL), and w = −0.99+0.11−0.13. The fR0 parameter is unconstrained in our fully non-linear f(R) cosmic shear analysis. Considering three
different model selection criteria, we find no clear preference for either the fiducial flat ΛCDM model or any of the considered extensions. In addition to extensions to the flat ΛCDM parameter space, we also explore restrictions to common subsets of the flat ΛCDM parameter space by fixing the amplitude of the primordial power spectrum to the Planck best-fit value, as well as adding external data from supernovae and lensing of the cosmic microwave background (CMB). Neither the beyond-ΛCDM models nor the imposed restrictions explored in this analysis are able to resolve the ∼ 3σ tension in S 8 between the 3×2pt constraints and the Planck temperature and polarisation data, with the exception of wCDM, where the S 8 tension is resolved. The tension in the wCDM case persists, however, when considering the joint S 8–w parameter space. The joint flat ΛCDM CMB lensing and 3 × 2pt analysis is found to yield tight constraints on Ωm = 0.307+0.008
−0.013, σ8 = 0.769+0.022−0.010, and S 8 = 0.779+0.013−0.013.