We present a theoretical analysis of electron heat flux inhibition in the solar wind when a significant portion of the heat flux is carried by strahl electrons. We adopt core-strahl velocity distribution functions typical for the solar wind at 0.3-4 au to demonstrate that strahl electrons are capable of generating highly oblique whistler waves at wave numbers k e ∼ 1, where e is typical thermal electron gyroradius. The whistler waves are driven by electrons in the anomalous cyclotron resonances (the fan instability) and propagate at typical angles of about 70°-80° to the strahl that is usually anti-sunward. The group velocity of the whistler waves is predominantly parallel to the strahl, thereby facilitating efficient scattering of strahl electrons. We suggest that the highly oblique whistler waves drive pitch-angle scattering of strahl electrons, resulting in halo formation and suppressing the heat flux of strahl electrons below a threshold that is shown to depend on βe. The proposed fan instability is fundamentally different from the whistler heat flux instability driven by the normal cyclotron resonance with halo electrons and being ineffective in suppressing the heat flux of the strahl.