Before clearing cookies -> close any opened windows\tab page with BBC iPlayer opened (if any)
Screenshots below shows exact steps.

Case 1:
When you’d like to delete cookies only for specific website.

Step 1: Choose Safari > Preferences, click “Privacy” tab and click “Manage Website Data…”

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Step 2:  Type  ‘bbc’ (1) and website’s list below will be filtered. Next click ‘Remove All’ (2) button.

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Step 3: You will be prompted with following dialog. Click ‘Remove now’ button

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Step 4: Click ‘Done’ button.

Case 2:
One step BUT it will delete ALL websites cookies and data – meaning ‘if you were logged in on another website, for example Facebook, then you will be logged off from it and you’ll have to log in again’.
If this is doesn’t matter for you – screenshot below shows “Two clicks cleaning”
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