Ouch! I flirted with tendonitis briefly early in my career, no fun. Here are 3 tips that worked for me:
1. Remember that electric bass guitar is one of the easiest
instruments to play. Very little force/tension should be required to play a properly-setup bass with good technique. Turn up your amp and play softer. Work with your teacher to refine every element of your technique until it is completely tension-free. If there is any pain then stop and evaluate why and fix it. If you have a chiropractor, then bring your bass to your next appointment and see if he/she can give suggestions about your posture. (If you don't then at least try practicing in front of a full length mirror.)
2. Avoid "spider" and "stretching" and "chops for chops sake" type exercises. My practice these days consists primarily of learning songs and common chord progressions, with a relaxed 1-2-4 left-hand fingering, moving up and down the neck freely. My right hand is also relaxed and explores the entire area between the bridge and the neck. The *worst* thing you can do is any type of exercise where one or both hands are locked in a fixed position for any length of time. Both hands should be flexing and expanding and contracting, moving around the instrument, so you don't get fatigue and cramps.
3. There are ways to practice without a bass in your hands. Sing along with the radio. Transcribe some tunes. Practice your music notation. Play some piano or guitar or bongos or trombone. Sit around with your band, listen to a recording of your last rehearsal, and talk through arrangements of the tunes. Watch a DVD of your favorite band for inspiration. I went through a period of about 3 years where I literally did not take my bass out of the case except for rehearsals and gigs with my band, and nobody complained that my playing had gone downhill. Visualize visualize visualize!
I hope that is helpful in some way, good luck!