Alice and Bob play the following game, taking turns. Alice starts and writes a non-zero single
digit number at the blackboard.
At every turn, each player adds a single digit at the right of the current number until a number
with 9 digits is reached.
If this number is divisible by 11, Alice wins otherwise Bob.
Note: This answer assumes that the non-zero restriction only holds for the first move, not for any subsequent digits, i.e. that the restriction was imposed only to ensure a valid 9-digit number was produced.
The winning player is
using the following strategy:
There is a well-known trick for reducing a number modulo 11, namely adding the digits at the odd positions and subtracting those at the even positions. In other words, Alice's moves minus Bob's moves results in a much smaller number with the same remainder mod 11 as the original nine-digit number.
Bob can do as his first move one less than Alice's digit, and in subsequent moves just copy Alice's digits. This ensures that the number is 1 modulo 11 after each of Bob's moves. Alice's last move can only result in a number that is 1 to 10 modulo 11, but not zero because that would require a digit of value 10.