In the framework of QFT, quantum fields are the fundamental objects instead of point-like notion of particles. Particles, at least fundamental ones like electron, are understood to arise as excitations of the quantum fields.

We know from experiments that a stationary free electron has some electric field associated with it. How does QFT explain that? Sure there is the background electromagnetic field and the electron field permeating all space-time but how does that explain electron itself generating electric field? In the QFT framework, it doesn't appear that electron can do that.

Can someone please help me with this? I have taken a QFT I course but this issue wasn't addressed there as well. Perhaps I am overlooking something very trivial.