Can we use ICA (Independent component analysis) on a noisy signal
Yes and no. 

The original ICA methods were not made for the case where there is noise on the signal. However, there exist relatively straightforward extensions that work for noisy signals (e.g. [*]). In this case, you can expect some degree of noise suppression but it won't be perfect. 

That means that ideally, your desired signal components will stick out enough from the noise for the ICA to recover them. Then again, you cannot expect this recovery to be perfect, there will be some estimation errors. The more noise, the worse the quality of your signal estimate. If you have weak signal components you should be prepared for the chance that they will become invisible under the noise floor. Give it a try I would say. 

[*] Hyvarinen, Aapo. "Gaussian moments for noisy independent component analysis." IEEE signal processing letters 6.6 (1999): 145-147.