She'll be standing on no legs if your dog does kill or injure her cat -she's aware of the risk, it is her legal responsibility to keep her pet on her property, it is not your responsibility to keep her pet out of yours.
However, you have to live next door to her, and legality aside, she can make life bloody awkward if she wants to.
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Find out who owns the fence between the properties - hopefully, it is yours.
Advise her to fit an inward overhang and roller for the most safe option - just to the adjoining side/end may do the trick. You can fit a roller to the top so her cat can't get a grip on the top of the fence, however if it gets in via some other route this WILL trap her cat IN your garden and she should be aware of that.
(If it isn't yours then its hers and theres not a lot more you can do there, but I would ask in writing (probably via text) so that you have a refusal in writing.)
Then should her cat get in, you've taken more than reasonable steps to stop it, make her aware of the options and risks or you have proof she refused to allow you to take reasonable steps to stop it.
Spend some time squirting her cat with a water pistol any time it comes over the fence. Owner won't like it but better a wet cat than a mullered cat.
The cat cut out 'ornaments' with reflective eyes can also help repel incoming cats however you'll need to teach your dog this is not a real cat... some will learn, some not so much!
One thing that will likely be suggested that I would NOT do, is muzzle your dog - a lurcher (and many other dogs) a/can grip fur/skin through a muzzle if they pin the prey between floor and muzzle and b/ will 'muzzle punch', effectively use their face/muzzle as a weapon and batter the prey to death. At the same time, this takes far longer than simply gripping small furry prey and shaking the head quickly so your dog is at risk of face/eye injuries from the frightened cat, which tend to further prolong an attack.