Without a large blackout I don't see how it should affect internet.
A totally isolated internet connection using fiber might be ok, but it still has a power feed using copper, so HV arcs getting into fiber routers could take them all out.
Some roadside comms cabinets are combined DSL and fiber. The DSL stuff is toast due to the voltage induced in all the copper phone wires and that may kill any connected fiber routers nearby. The main issue is you can't just install a replacement because everyone else is trying to do the same at the same time. There's just not enough replacements available. So it would take months, maybe years, maybe 10's of years to manufacture enough replacement electronics depending how bad it was.
The carrington event had reports of sparks showering from telegraph machines, shocking operators and setting papers ablaze. The arcs were pretty extreme and had a lot more energy than a static shock.
The carrington event is not even the biggest, just the biggest in recorded history. There's some evidence in tree rings of an even large one happening around 14300 years ago.
It really just comes down to how big it is, small ones happen all the time and don't case many issues, they're a bit annoying for power stations to deal with but no big issue normally.
However really big ones have the power to fry a lot of modern electronics if those systems are connected to any long conductive wires.
It's like taking a car ignition coil to your tech.
Its not going to be happy.