Diane, I feel rather silly. I spent a lot of time scouring the registers for this "Cliff" family only to discover that the surname is definitely "Oliff" (or Olive in modern terms). In almost every instance what I took as a C was a fully closed loop (or "O"), often with a strike through. In any event, since these might be of interest to you I recorded what I found. FYI, I searched baptisms from 1559 to 1640 and marriages from 1571 to 1618. Note that many of the register entries were in latin (which I do not know) and the writing was often very difficult, so take this with a grain of salt. The following entries may be related to this family:

21 Jan 1578 Willm Ollife married Jone Luckyn
2 Jan 1581/2 Agnes dtr of Wm ?Lambat baptized
27 Jan 1587/8 Wm son of Wm Lambert baptized
24 Nov 1589 Ellen Oliffe baptized
10 Aug 1596 ?Rednes Olyffe als Lambard son of Wm baptized
12 Aug 1605 Thomas Seman and Agness Oliff als Lambert married
11 Jul 1611 Tomas Oliff als Lambard and Johanna Barker married
23 Aug 1613 Wm Olyff als Lambard son of Wm and Johan baptized
26 Jan 1614 Robert Kettell and Helen Oliff als Lambart married
Sep 1615 Christian Lambert als Olyff dtr of Thomas and Johan baptized
31 May 1620 ____ of Thomas Olyff als Lambert, sailor and Jone baptized
30 Nov 1617 William Oliff als Lambard and Priscilla ?Shennyng married
Aug 1620 Wm Olyffe son of Wm Olyff als ?Lambard saylor and Priscilla baptized
Aug 1621 James Olyff als Lambert(?d) son of Thomas Olyff, mariner and Joane baptized
Aug 1625 Grace Olliff als Lambart dtr of Tho ?Olyfft als Lambert, mariner and Joane baptized
7 Sep 1628 Grace Olliff als Lambert dtr of Thomas Olliff als Lambert, mariner and Joane baptized
7 Sep 1632 Anne Ollyff als Lambert dtr of Wm Ollyff als Lambert, mariner and Anne baptized
29 Jan 1636/7 Johanes of William Oliff als Lambert and Judith baptized

Hope this is of interest to you. Bill