This paper reports the production of a cellulase-free and alkali-stable xylanase in high titre from a newly isolated Bacillus pumilus SV-85S using cheap and easily available agro-residue wheat bran. Optimization of fermentation conditions enhanced the enzyme production to 2995.20 ± 200.00 IU/ml, which was 9.91-fold higher than the activity under unoptimized basal medium (302.2 IU/ml). Statistical optimization using response-surface methodology was employed to obtain a cumulative effect of peptone, yeast extract, and potassium nitrate (KNO₃) on enzyme production. A 2³ central composite design best optimized the nitrogen source at the 0 level for peptone and yeast extract and at the −α level for KNO₃, along with 5.38-fold increase in xylanase activity. Addition of 0.1% tween 80 to the medium increased production by 1.5-fold. Optimum pH for xylanase was 6.0. The enzyme was 100% stable over the pH range from 5 to 11 for 1 h at 37°C and it lost no activity, even after 3 h of incubation at pH 7, 8, and 9. Optimum temperature for the enzyme was 50°C, but the enzyme displayed 78% residual activity even at 65°C. The enzyme retained 50% activity after an incubation of 1 h at 60°C. Characteristics of B. pumilus SV-85S xylanase, including its cellulase-free nature, stability in alkali over a long duration, along with high-level production, are particularly suited to the paper and pulp industry.