Let F be a finite field with q elements and characteristic 3. A sum
M = M_{1}^4+\ldots+ M_{s}^4
of fourth powers of polynomials M_1,\dots, M_{s} is a strict one if 4\deg M_i < 4 + \deg M for each i= 1,\ldots, s. Our main results are: Let P\in F[T] of degree \geq 329. If q>81 is congruent to 1 (mod. 4), then P is the strict sum of 9 fourth powers; if q=81 or if q>3 is congruent to 3 (mod 4), then P is the strict sum of 10 fourth powers. If q=3, every P\in F[T] which is a sum of fourth powers is a strict sum of 12 fourth powers, if
q=9, every P\in F[T] which is a sum of fourth powers and whose degree is not divisible by 4 is a strict sum of 8 fourth powers; every P\in F[T] which is a sum of fourth powers, whose degree is divisible by 4 and whose leading coefficient is a fourth power is a strict sum of 7 fourth powers.