I haven't had a PICC in years (my veins became to sclerosed, so I was no longer a candidate for it), but when I did have it, I didn't let it slow me down much at all. I limited the amount of lifting I did with it, and I wrapped an ACE wrap around it to help stabilize it at the elbow, but other than that I did everything else. Everything else included:
showering (with it covered in saran wrap)
riding roller coasters
being in the school musical (lots of activity and dancing)
playing field hockey
twirling baton (I was a majorette in high school)
Yes, you should be careful, but don't let it run your life. If you can, request that they place it in your non-dominant arm. I am right handed most of the time, so I had the PICC placed in the left arm.
Good luck!