There are a reasonable number of MPs who support compromise on free movement, i.e. leave the EU but reach an agreement with the EU similar to the ones that non-EU members Norway, Iceland, and Switzerland have signed. That's one of the "soft Brexit" options. It's not guaranteed that the EU would accept this proposal, but it's something the British government could choose to seek.
Whether this is consistent with the referendum result is arguable either way I think. The majority of pro-exit campaigners, and most likely of Leave voters, were against free movement, with regaining exclusive UK control over immigration being one of the foremost demands. So a soft Brexit that retains free movement could be seen as not following the spirit of the outcome, even if it follows the letter of it (by formally leaving the EU). On the other hand, the Leave majority was quite slim, and a subset of the Leave voters, probably at least 10%, in fact did think they were voting for a soft exit to more of a Switzerland-style status. So it is probably the case that while Brexit, when asked about in general terms, had 52% support, if you explicitly specify "hard Brexit" including exit from the common market, support would drop. But none of this was made precise in the referendum question, so what the referendum outcome gave a mandate for is what's now being debated.