I was curious about surveys that examine reasons for the Brexit and what different factors determined individual votes.
In specific, was rising Islam in Europe one of the factors without which the referendum result would have been different?
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There were plenty of polls which asked about peoples' reason for voting to leave. While the wider issue of immigration was an option on these polls, I'm not aware of any that asked mentioned Islam specifically. And while there will be some Leave voters who voted with this motivation (one example reported in HuffPost), it's likely that this person was part of a very small minority.
According to this Survation poll, conducted the week of the referendum, the top reasons for Leave voters deciding to vote that way are:
While immigration was the top issue, the overwhelming share of that concern was about low-skilled immigration from Eastern Europe, not about Muslim immigration.