I'm not saying that 52% of the UK voted leave because of immigration, but I'm certain that at least 4% did, which is enough to clinch the win. In an otherwise tight race, that is a big difference so I think it really has trumped other policy issues in this case. But I can't 'prove' it if that's what you're looking for. I don't think it is an unreasonable statement given that the polls repeatedly identified immigration as the key issue driving the leave vote eg [1].

[1] http://m.smh.com.au/world/brexit-poll-why-the-british-people...