No, they are actually slow (the pulses of them bumping each other in sequence travels fast, not the electrons themselves). They do experience Lorentz force though, which arises when electrical charge moves inside of a magnetic field. This force wants to make electrons turn sideways from a straight path, trying to make them circle around perpendicular to the magnetic field.
The magnetic field of wires causing Lorentz force stress on other wires inside microchips is one of the ways they can really slowly wear out (note: we are talking about a decades/centuries timeframe).
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