Walker lost by 2 and 1 to England's Danielle Montgomery, while Caithness went out at the final hole against another English player, Felicity Johnson. Caithness was one up with six to play but lost the 13th and 14th before bouncing back to level the match at the 15th.
But a win for Johnson at the par-four 17th put her one up and she held on to secure the one-hole victory and a second-round match against Wales' Lydia Hall.
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Hide AdEngland's Laura Davies battled to a 2 and 1 win over American Hannah Jun to set up an intriguing last-16 match against Wales' Becky Brewerton.