Opened up the YouTube app and was greeted with this offer. Available for YouTube premium and Google Play Music subscribers.

Try your luck with this old link. Several users have reported success. (Thanks to R3XNebular.)

Terms and conditions

Promotional code offer is provided by Google Australia Pty Ltd and subject to the following terms. Offer must be redeemed by 6.59 p.m. AEDT on 24 January 2020 or it will expire. Available only to users who are current paid members of individual plan, student plan or family plan, including head of household and dependants, of YouTube Premium, YouTube Music Premium or Google Play Music on 4 December 2019. User's membership must not be in a paused state. Each promotional code is valid for one redemption of a Google Nest Mini and will be available on a first-come, first-served basis, subject to availability. Google reserves the right to modify these terms as needed.

To redeem your promotional code, visit Google Store URL and enter the promotional code during checkout to receive a discount in the amount of the retail price plus taxes, if applicable, on a single Google Nest Mini. Promotional code discount will be applied at checkout. If you access the Google Store by clicking on the redemption email or in-app notification that we send to you, the promotional code will be applied automatically during checkout. Promotional code must be redeemed by 6.59 p.m. AEDT on 24 January 2020 or it will expire. Promotional codes cannot be applied to previous purchases and cannot be combined with other offers. Subject to availability. Credit card entry may be required for redemption. Google Nest Mini can only be redeemed and delivered to Canada, Australia, Japan and parts of Europe. Delivery and handling costs are included. Google Store Promotion Code Terms and Conditions apply.

Update: Australian account required. Users paying in overseas currencies don't appear to be eligible.