Are we more productive under deadline or when there’s no time pressure?

Neither.

Heavy time pressure kills creativity and morale. On the other hand, having no deadlines was not optimal either. Much like with stress, low-to-moderate time pressure produces the best results.

Via The Progress Principle: Using Small Wins to Ignite Joy, Engagement, and Creativity at Work:

Time pressure is one of the most interesting forces we studied. Although occasional time pressure for short periods can be exhilarating, using extreme time-pressure to stimulate positive inner work life, for weeks on end or even in the short run, is playing with fire (see âTime Pressure and Creativityâ). If managers regularly set impossibly short time-frames or impossibly high workloads, employees become stressed, unhappy, and unmotivatedâburned out. Yet, people hate being bored. it was rare for any participant in our study to report a day with very low time pressure, such daysâwhen they did occurâwere also not conducive to positive inner work life. In general, then, low-to-moderate time pressure seems optimal for sustaining positive thoughts, feelings, and drives.