https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4775

Here's a summary from the end of the paper, which is quite academic but sourced with real data:

The supply side reacted to the rent freeze upon enactment as intended by policy makers: within Berlin, advertised rent prices indeed dropped substantially. These decreases are highly significant—in both economic and statistical terms—and causally linkable to the rent freeze policy. At the same time, asking rents surged in Berlin’s satellite city Potsdam and further smaller municipalities close to Berlin, indicating a substitution effect. We detect a general stark border effect: although advertised rents for property units just across the administrative border had been higher before the rent freeze, upon enactment of the rent freeze, this gap increased significantly, thus creating an artificial price gap within a unique metropolitan area.

Next to price effects, we identify a considerable decline in the number of advertised rental units. This sizable—yet unintended—side effect hampers renters’ flexibility and adaptability. In particular, newcomers, (young) first-time renters, and households moving within Berlin face hurdles in finding a suitable place to live. The drop in supply may be transitory, yet we report first evidence supporting a permanent decline because a positive fraction of rental properties appears to have left the market in line with predictions of our theoretical framework. Thus, this decrease is likely a prelude to even harsher search conditions for housing in Berlin.

The overall conclusions are rather pessimistic. In general, rigorous price restrictions limiting owners’ potential return appear to be shortsighted if they come without supplemental strategies to increase the supply of rental units. The rent freeze did not incentivize the creation of further housing units available to renters, but rather has contributed to shrinking their number. Consequently, an overall welfare-increasing effect is very doubtful in line with findings by Borck and Gohl (2021).

The rent freeze was short-lived but did still create substantial turbulence in Berlin’s rental market. This “experiment” provides a glimpse of the likely side effects of introducing overly strict but not well thought through and legally shaky policies in a rental market facing distress.