Fission-fragment mass and total-kinetic-energy (TKE) distributions following fission of even-even nuclides in the region 74 ≤ Z≤ 126 and 92 ≤ N≤ 230 , comprising 896 nuclides have been calculated using the Brownian shape-motion method. The emphasis is the region of superheavy nuclei. To show compatibility with earlier results the calculations are extended to include earlier studied regions. An island of asymmetric fission is obtained in the superheavy region, 106 ≤ Z≤ 114 and 162 ≤ N≤ 176 , where the heavy fragment is found to be close to 208Pb and the light fragment adjusts accordingly. Most experimentally observed α-decay chains of superheavy nuclei with Z> 113 terminate by spontaneous fission in our predicted region of asymmetric fission. In these cases, the pronounced large asymmetry is accompanied by a low TKE value compatible with measurements.