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A Hamiltonian six-field gyrofluid model is constructed, based on closure relations derived from the so-called "quasi-static" gyrokinetic linear theory where the fields are assumed to propagate with a parallel phase velocity much smaller than the parallel particle thermal velocities. The main properties of this model, primarily aimed at exploring basic phenomena of interest for space plasmas such as the solar wind, are its ability to provide a reasonable agreement with kinetic theory for linear low-frequency modes, and at the same time to ensure a Hamiltonian structure in the absence of explicit dissipation. The model accounts for equilibrium temperature anisotropy, ion and electron finite Larmor radius corrections, electron inertia, magnetic fluctuations along the direction of a strong guide field, and parallel Landau damping. Remarkably, the quasi-static closure leads to exact expressions for the nonlinear terms involving gyroaveraged electromagnetic perturbations. One of the consequences is that a rather natural identification of the Hamiltonian structure of the model becomes possible when Landau damping is neglected. A slight variant of the model consists of a four-field Hamiltonian reduction of the original six-field model, which we use for the subsequent linear analysis. In the latter, the dispersion relations of kinetic Alfvén waves and the firehose instability are shown to be correctly reproduced, relatively far in the sub-ion range (depending on the plasma parameters), while the spectral range where the slow-wave dispersion relation and the field-swelling instabilities are precisely described is less extended. This loss of accuracy originates from the breaking of the condition of small phase velocity, relative to the parallel thermal velocity of the electrons (for kinetic Alfvén waves and firehose instability) or of the ions (in the case of the field-swelling instabilities).