This is called “shopping the deal” and it appeals to a certain type of customer but the short answer is no, we won’t do that the majority of the time.
Each bank runs a Credit Check on you when they approve you and the other banks can see this. This not only lowers your Credit Score (which is not great) but when the bank comes to offer you interest rates, they are not going to bother fighting for a good deal because you are clearly looking at other competitors.
This is counter-intuitive to what you might think. Often people think that banks realize they’re in competition and sharpen their pencils. Branch staff will tell you that competition is healthy, but trust us, the internal staff respond much better to loyal customers.