Hi Federico Fiorini,
It should be your code's problem.
Please run my attached LPC11C24 project which is based on the LPCXpresso IDE or the MCUXpresso IDE.
I have run it on my LPCXpresso11C24 board, it works OK.
The code will run in the deep sleep mode, P1.4 pin is the wake up pin. After you download my code to your LPC11C24, you need to connect P1.4 pin to 3.3V, then reset the chip, you will find LED is not ON, because the chip is enter the deep sleep mode, after you connect P1.4 pin to low, you will find the LED is ON, it means your MCU is waked up.
Please note, in the low power mode, debugger will be disconnected, so you need to download the code to the MCU to test without debug.
Please try it on your side.
You can find the LPCXpresso IDE or the MCUXpress IDE from this link:
1. LPCXpresso IDE
LPCXpresso IDE v8.2.2|NXP
This is the old IDE, now can be replaced by the MCUXpresso IDE, and the LPCXpresso IDE project can run in the MCUXpresso IDE directly.
2. MCUXpresso IDE
MCUXpresso IDE|Eclipse-based Integrated Development Environment (IDE)|NXP
Please note, when you use the MCUXPresso IDE to import the LPCXpresso IDE, please copy the project to the workspace.
Just like this:

Wish it helps you!
If you still have question about it, please kindly let me know!
Have a great day,
Kerry
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