Do post-exercise ice baths improve performance?

There is no direct yes, or no answer and importantly it depends on the context and what type of performance that is considered. Nevertheless, when trying to summaries recent studies it appears as, if performance would be considered improved by reducing the symptoms of exercise-induced delayed onset muscle soreness (DOMS, i.e., pain and stiffness experienced in muscles several hours to days (usually 24-72 hours) after unaccustomed or strenuous exercise), then there are reports supporting that post-exercise ice baths have an acute alleviating effect. The physiological explanation is likely linked to that cooling i) reduces nerve impulse transmission and thus reduces the level of pain perception and ii) induces constriction of blood vessels in peripheral tissue (e.g., muscle) which results in reduced fluid diffusion that may assist in reducing exercise-induced acute inflammation and hence reduce pain, swelling and the loss of force associated with inflammation.  

On the other hand, if performance is measured as one-repetition maximum, maximum isometric strength, or strength endurance performance then post-exercise ice baths appear to induce deleterious effects (Malta et al 2021; Petersen Fyfe 2021). Moreover, if performance is measure as acute recovery of exercise performance following fatigue-induced endurance exercise then cooling also has a negative impact (Cheng et al 2017). Cooling results in reduced skeletal muscle blood flow and nutrient delivery to the muscle, which may contribute to the impaired strength and anabolic response caused by cooling during recovery from resistance exercise. Furthermore, glycogen is a primary energy source for muscle contraction, and depletion of glycogen in muscle is linked to prolonged force depression after fatiguing exercise. Cooling is shown to reduce glycogen resynthesis rates and hence lowering muscle temperature results in impaired recovery from exhaustive endurance exercise (Cheng et al 2017).

Thus, unless improve performance is associated with an acute (i.e short-term) alleviation of DOMS, it is unlikely that post-exercise ice baths have beneficial effects. Instead, they may even induce detrimental effects on performance associated with muscle function.