Can you stop SQL for a moment and recheck ping? with SQL stopped ping should go back to normal if SQL is the problem - this will avoid driver updates I think.
do something DNS checks too - be surprised how many things need good DNS lookups to work properly - if a SQL job is trying to authenticate or find a remote server it could be causing a problem.
Also look in the event log - any issues around disk drives? a disk on the way to failing could make many such issues. if SQL is unable to reliably write a transaction log it's going to cause hell.