Index to the wills proved in the Consistory court of Carlisle, 1661-1750 (1998); published by British Record Society lists the following wills under 'Barton' and 'Burton' (the two names sometimes seem to get confused):
W = will, W[N] = nuncapative will, A = administration bond (probate), I = inventory

Documents that have been acquired from *Cumbria archive service are underlined below (transcripts of some, but not yet all, can be found on this site).

p20

• Rob. Barton, Dalston (Ivegill). 1662 AI [54]
• John Barton, Dalston (Ivegill). 1662 AI [60]
• Rob. Barton, Shap (Hardendale Nab) [We]. 1671 I
• Eliz. Barton, Kirkby Stephen (Winton) [We]. 1687 AI
• Bernard Barton, Dalston (Ivegill). 1715 W (transcript here)
• John Barton, yeo., Hesket-in-the-Forest (Ivegill Head). 1720 WI (transcript here)
• And. Barton, tanner, Penrith. 1722 WI
• John Barton, yeo., Dalston (Ivegill). 1747 WI 336 [183r] (transcript here)
• Ann Barton, wid., Dalston (Ivegill). 1749 WI 5 [12v] (transcript here)

pp51-52

• Ann Burton, wid., Long Marton (Brampton) [We]. 1690 AI
• Anth. Burton, yeo., Dalston. 1740 WI 201 [80v]
• Chris. Burton, Skelton (Unthank). 1672 AI
• Francis Burton, yeo., Shap (Hegdale) [We]. 1721 WI
• Geo. Burton, Brough [We]. 1661 W [28]
• Jas. Burton, Greystoke (Watermillock, Baldhow). 1746 WI 19 [170r]
• John Burton, yeo., Kirkandrews-on-Eden. 1733 AI [137]
• John Burton, sen., yeo., Penrith. 1735 WI 24 [204]
• Oswald Burton, ?Shap (Boat Howe) [We]. 1672 I
• Rob. Burton, yeo., Penrith (Townhead). 1688 WI
• Wm. Burton, Crosthwaite (Underskiddaw, Applethwaite). 1662 WI [77]
• Wm. Burton alias Marti, yeo., Greystoke (Stoddah). 1667 WI [173]
• Wm. Burton, Kirkandrews-on-Esk. 1681 WI
• Wm. Burton, yeo., Penrith. 1689 WI
• Wm. Burton, sen., yeo., Kirkandrews-on-Eden. 1735 WI 13 [187]
• Wm. Burton, clerk, Kirkby Thore [We]. 1746 W 15 [165r]

Unlisted

The *Cumbria archive service has also provided the following wills not listed in the index above:

• Robert Barton (Dalston). 1588 AI
• Barnard Barton (Dalston). 1633 W (transcript here)
• George Barton (Ivegill, Dalston). 1630 W[N] (transcript here)
• John Barton. 1617 I
• Mabel Barton (proved Carlisle). 1639 W[N] (transcript here)