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We present a three-dimensional computational study of the impact of external magnetic fields on the dynamics of superparamagnetic ferrofluid droplets and rheology of dilute ferrofluid emulsions in planar extensional flows. Specifically, we show how the intensity and direction of uniform magnetic fields affect the planar extensional rheology of ferrofluid emulsions by changing the shape and magnetization of the constituent ferrofluid droplets in suspension. We find that the two traditional extensional viscosities associated with the normal stresses of the bulk emulsion in extension either remain constant or increase with the field intensity; the only exception occurs when the field direction is perpendicular to the extension plane, where increasing the field intensity keeps the planar extensional viscosity constant and modestly decreases the second extensional viscosity. We also find that the droplet tilts in the flow when the external field is not aligned with one of the flow main directions, which changes the recirculation pattern and flow topology inside the droplet. At the microscopic level, the droplet experiences a magnetic torque because of a small misalignment between its magnetization and the external field direction. At the macroscopic level, the bulk emulsion experiences a field-induced internal torque that leads to a nonsymmetric stress tensor with unexpected shear components in extension. To account for this unconventional stress-strain response, we introduce new extensional material functions such as shear and rotational viscosity coefficients that unveil novel rheological signatures of ferrofluid emulsions in planar extensional flows. This study offers new insights into applications based on the field-assisted manipulation of ferrofluid droplets and sheds light on the potential of ferrofluid emulsions as a model system for chiral fluids with internal rotational degrees of freedom that can be activated and controlled by coupling static magnetic fields with hydrodynamic flows.