I have worked out pretty much every second to third day (more than every 2 days is useless, see
Overtraining) for quite some time (~2
1/2 - 3 years) now, with some short pauses inbetween. I started even before playing piano, and it didn't affect me in any way.
Even when I'm taking longer breaks (as right now, 2 weeks already, muscles take about 1 week to recover 100% from a hard training), my technique doesn't suddenly improve.
Just for your information, I play for example the Rach 23/5, so I do need at least some level of technique, which, as far as I can tell, is not influenced in a bad way by physical workout.
I think your teacher is just cautious because s/he doesn't really know wheter or not it will affect your playing or not, so s/he just automatically says no. But that's BS.
Also your fingers don't get fatter by working out, they do if you eat too much.
Again, from experience I can say: my fingers didn't get bigger from working out (they were just big to begin with lol) and neither will yours.
So my conclusion:
I don't know what happens when you try to become Arnold Schwarzenegger, but normal workout doesn't do a thing to your piano skills.
tl;dr: Yes.