Is my pump faulty or my tyre?
As other have commented since you can fill the tires with another pump the issue is not your tires. If you bought it at your LBS (local bike shop) I would bring it in and have them demonstrate its' proper use. If they  can't get it to work they should replace it at no charge. While the listed capacity is 160psi, the amount of effort required to generate the pressure will vary with pump design. A larger diameter pump will require fewer pumps to inflate a tire but with more effort. While not as much of an issue with a floor pump the difference can be considerable with a small pocket pump.