Is there any way to open it without causing any damages?
This type of lock is very basic. They are not intended to be really secure, just so that the casual snooper or thief can't open it easily.
That is why they are easy to pick with a bent wire, such as a reshaped paper clip. You might only need to feel for a simple small lever inside, to flick across. You should do this with the right amount of pressure on the outer hasp, to remove pressure from the catch.
Alternatively, if you have others keys around, try them.
Old luggage locks like this are usually "warded locks", a very simple design with very little security. They can be opened with a pretty simple tool which you can probably make out of junk if you need to, or you can obtain a set of warded lock-picks on Amazon for very little outlay. (Other sources are doubtless available.)
Here's a demonstration by a well-known Youtube lockpicker.
Newer luggage locks are often of a type that can be opened with a TSA key, recognisable through a red diamond-shaped logo and a TSA00X code printed next to the keyhole. Complete sets of TSA keys are also readily available by mail-order. These locks are usually also quite easy to pick using raking tools.
Ideally, use the proper key.
Alternatively, use the services of some local company specialized in locks and doors, to open the lock for you. They have the required generic tools and the practical experience to do it for you. (it is pretty much what you see in the movies, using some bent wires to unlock locked doors)