Why, make a costume as an excuse, of course. The only other character I could think of who wore an extremely long scarf was Rikku from FFX. Conveniently, FFX-2 explained her outfit away as proper Thief wear, so it was relatively easy to substitute the Doctor's scarf instead. Shortly thereafter I made a Doctor-themed headband and adapted Doctor-themed sleeves, too.
This is the pattern for the sleeves. I left them in garter stitch to match the scarf, but feel free to mess with it all you like. This is more of a basic guide than a true pattern and I welcome you to edit it to suite your needs.
The yarn I used was Vanna's Choice and I knit with a size 9 pair of circs, but I'm a loose knitter, I think. Without stretching, these wrap around my arm and leave only a 1-1.5 inch gap for the lacing. If the gap is too large or small for your liking, add or remove stitches to change the size.
At row 32, the sleeve reached from my upper arm to my elbow. If you are taller or shorter than me, add more or decrease the rows before you start the increasing rows.
Cast on 35.
Row 1: Knit across
Row 2: Knit across
Row 3: K1, yo, k2tog, K to last 3 stitches, k2tog, yo, K1
Rows 4-8: Knit across
Repeat rows 1-8 3 more times.
The pattern should make a solid row, a hole row, 2 solid rows, repeat for a total of 4 sets, 16 garter rows, 32 rows. Then we allow it to increase.
Row 33: Knit across
Row 34: Knit across
Row 35: K1, yo, K to last stitch, yo, K1
Row 36-40: Knit across
Repeat Rows 33-40 until you reach row 108, or your own desired length.
Bind off, weave in ends.
To finish, you can use a bulky yarn or a ribbon to lace the sleeves. I used a lucet and the beige Vanna's Choice to make a cord for mine (Google Lucet, if you don't know).
To help keep them on my arms, I cut a strip of fabric and hand-sewed it into the top of the sleeves, and when I wore them, I used fashion tape to adhere them in place.
Good luck and have fun!