Not BA, but I had an interesting experience on Easyjet with this. Had gone through process of booking 2 seats EDI-TFS, got quoted price, but by time I entered my credit card details got a message the fare was no longer available and I needed to pay an extra £200. Went from Firefox to Chrome and was then quoted my original fare cost!
Not sure if this was just LCC sharp practice or based on browsing history as I had been keeping an eye on the route/price in Firefox.