Electromagnetic instabilities and turbulence driven by the electron-temperature gradient are considered in a local slab model of a tokamak-like plasma, with constant equilibrium gradients (including magnetic drifts but no magnetic shear). The model describes perturbations at scales both larger and smaller than the electron inertial scale d_e, at which flux unfreezes, and so captures both electrostatic and EM regimes of turbulence. The well-known electrostatic instabilities — slab and curvature-mediated ETG — are recovered, and a new instability is found in the electromagnetic regime, called the Thermo-Alfvénic instability (TAI). It exists in both a slab version (destabilising kinetic Alfvén waves) and a curvature-mediated version, which is a cousin of the (electron-scale version of) the kinetic ballooning mode (KBM). The curvature-mediated TAI turns out to be dominant at the largest scales covered by the model (greater than $d_e$, but smaller than the ion gyroradius), trumping curvature-mediated ETG and exciting electromagnetic perturbations with a specific parallel wavenumber (unlike the curvature-mediated ETG, which is two-dimensional). Its physical mechanism hinges on the fast equalisation of the total temperature along perturbed magnetic field lines (in contrast to the KBM, which is approximately pressure balanced). It turns out that it is then possible to construct a turbulent cascade theory with two energy-injection scales: d_e, where the drivers are slab ETG and slab TAI, and a larger scale dependent on the parallel size of the system, where the driver is curvature-mediated TAI. The latter dominates the turbulent transport if the temperature gradient is greater than a certain critical value, which scales inversely with the electron beta. The resulting heat flux scales more steeply with the temperature gradient than the heat flux due to electrostatic ETG turbulence in this regime, and thus gives rise to stiffer transport. This can be viewed as a physical argument in favour of near-marginal steady-state in electron-transport-controlled plasmas (e.g., the pedestal). While the model is simplistic, the new physics that is revealed by it should be of some concern, or at least interest, to those attempting to model the effect of electromagnetic turbulence in tokamak-relevant configurations with high beta and large electron temperature gradients.