Are there rodents (such as the house mouse) in Antarctica?

Yes according to this article from December 2014:

The biggest island-based rat-killing operation in history is under way on South Georgia, an island north of Antarctica and east of the Falklands. The island was once one of the richest seabird breeding territories in the world, but bird populations fell into severe decline after rats arrived aboard sealing and whaling ships in the 19th century. Scientists are racing to eradicate the rats before glaciers that currently serve as natural barriers between the rodents and remaining bird populations melt away. They are hoping to save the South Georgia Pipit, a species found nowhere else in the world, from extinction.

http://earthsky.org/earth/killing-rats-to-save-birds-as-glaciers-recede

No, there are no rodents that live in Antarctica.
Some mammals that do live in Antarctica include a number of different species of seals and whales.

https://en.wikipedia.org/wiki/List_of_mammals_of_Antarctica

The answer comes down to a technicality and the convoluted rules for determining which islands are officially attached to Antarctica. If the intent was to cover the mainland Antarctic continent only, then NO is completely correct. If not then you must first determine what constitutes the boundary of Antarctica as it extends out to sea, and which islands are Antarctic and which are not.

There are two geographical limits at play there, the very well defined and static line of 60 degrees south latitude. This is the northward boundary of the area under the jurisdiction of the Antarctic Treaty. Islands south of this line cannot be claimed as the territory of a single nation. The other boundary at play is the Antarctic Convergence. This represents the area where cold northbound Antarctic water meets "warm" subantarctic waters to the north. Though it is somewhat fixed in location, it can move over time and is far from a perfectly symmetrical circle. The Convergence falls further South than 60 degrees south in some ranges but generally occurs north of that line as you trace it around the globe. Some Conventions use this Convergence as the northern limit of Antarctic territory which creates a grey area in the range between the two lines where islands may or may not be considered part of Antarctica.

South Georgia Island falls in this slip of Ocean. The second link suggests the island should be considered sub Antarctic.

https://en.m.wikipedia.org/wiki/Antarctic_Convergence
https://en.m.wikipedia.org/wiki/Subantarctic