We present constraints on extensions to the flat $Lambda$CDM cosmological model by varying the spatial curvature $Omega_K$, the sum of the neutrino masses $sum m_ u$, the dark energy equation of state parameter $w$, and the Hu-Sawicki $f(R)$ gravity $f_{R0}$ parameter. With the combined $3times2$pt 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 $Lambda$CDM model with $Omega_K=0.011^{+0.054}_{-0.057}$, $sum m_ u<1.76$ eV (95% CL), and $w=-0.99^{+0.11}_{-0.13}$. The $f_{R0}$ 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 $Lambda$CDM model or any of the considered extensions. Besides extensions to the flat $Lambda$CDM parameter space, we also explore restrictions to common subsets of the flat $Lambda$CDM 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 CMB. Neither the beyond-$Lambda$CDM models nor the imposed restrictions explored in this analysis are able to resolve the $sim 3sigma$ tension in $S_8$ between the $3times2$pt constraints and Planck, with the exception of $w$CDM, where the $S_8$ tension is resolved. The tension in the $w$CDM case persists, however, when considering the joint $S_8$-$w$ parameter space. The joint flat $Lambda$CDM CMB lensing and $3times2$pt analysis is found to yield tight constraints on $Omega_{rm m}=0.307^{+0.008}_{-0.013}$, $sigma_8=0.769^{+0.022}_{-0.010}$, and $S_8=0.779^{+0.013}_{-0.013}$.
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