It's just the day after Christmas, but Amazon is already boasting about a successful holiday season, calling 2017 the "biggest" ever.

The e-commerce giant said Tuesday morning that customers were shopping on its website at "record levels" throughout November and December, and over the course of one week, more than 4 million people became Prime members or started a free trial, sending Amazon's membership count to new highs. However, the company didn't provide a specific membership total.

Meantime, Amazon said its devices also logged their best holiday, with the Echo Dot and Fire TV Stick leading the way. The Amazon Alexa app, which helps set up the company's voice-enabled devices to Wi-Fi, was a top downloaded application via Apple over the holiday weekend.

The black and white versions of the Echo Dot (on sale for $29.99) were such a popular gift that as of Tuesday, they weren't available to be shipped from Amazon until early January.

The company also said shopping on its Amazon app increased nearly 70 percent this holiday season, as more customers become comfortable using their mobile devices to make purchases. Millions of tech gadgets were ordered from Amazon this way, the company added.

Other best-selling items through Christmas included the Instant Pot multicooker, WowWee Fingerlings toys, the Fitbit Charge 2 fitness tracker, a 23andMe DNA test and robot vacuums for cleaning around the house.

This holiday season, Amazon is again expected to grab the bulk, or more than 50 percent, of new digital sales. That leaves retailers like Walmart, Target, Kohl's and Best Buy thinking of new ways to either draw shoppers into stores or win dollars online.

In turn, Amazon has been looking to open more physical locations across the U.S. to make pickups and returns easier. Growing its private-label lines and partnering with other big-name retailers also remain in focus in the coming year.