Peter here from Shopify Support.
While it would be possible to create an ongoing discount code for a specific customer, it would still require them to manually enter the code at checkout as it can't be applied automatically. To get this setup, you'll want to navigate to your discounts area and select "create discount."
From here, you'll give the discount code a name, enter in the discount type and percentage, and who the discount applies to:
You could choose to limit the number of times the customer can use the discount code, however, you won't want to limit it to one per customer as this would defeat the purpose of an ongoing discount code. Once you've entered in the above information, you can click save and that should create the discount you're after.
One thing to note is that this discount can't be stacked with other discount codes, so your customer would need to pick which discount they'd prefer to use. Another thing you could consider is using Locksmith to protect a discounted product. This way, only specific customers can reach the discounted product page, while still being able to use other discounts at checkout.
If there's anything else I can help with please don't hesitate to reply back and I'll happily help out!
All the best,