Thank you for responding,
This is not my family and I have no actual findings to share with you, but have you tried the free access resource at Family Search? https://familysearch.org/ark:/61903/3:1:S3HY-6P44-SHF?mode=g...
has images of original Binham PR covering "Baptisms, Marriages, Burials 1559-1653". Some of these images are particularly difficult to read but if you find them impossible you may find something useful in the alternative (but incomplete) Bishops and Achdeacons' transcript image collections for Norfolk, also available on the Family Search website.
In any case, individual images can be enlarged within the Family Search website view; and also downloaded, offering the possibility of manipulating contrast & brightness on your own computer to aid reading.
If you have any doubt about years of birth or christening you could search over several years and of course allow for name variations, but you will know that.
Particularly in view of the religeous upsets at about that period, and if you have not done so already, you may also wish to search the surviving non-conformist records as well as those of the established church.
Your reference to Binnham may come from the same source as two 19/01/2011 submissions for a William Beck b1635 in the 'IGI', now archived within Family Search. perhaps curious that both were posted on the same date but maybe one or other of the numbered subscribers can be contacted and may now know more.