Is it permissible to go on a mixed-gender trip—originally not mixed-gender—with the intention of strengthening relationships and changing the work atmosphere, knowing that the majority of participants are people of faith and good character?

It is not permissible for a woman to travel without a mahram (definition), whether the journey is short or long, because the Prophet, peace and blessings be upon him, said: "A woman should not travel except with a mahram." Furthermore, journeys where non-related men and women mix are a gateway to fitna (temptation/strife) and a means to corruption.