Hi Daniel,
The TPS54160A will be able to handle the application. The condition which may cause an issue is with a 20mA load and 14V input. The boot UVLO may trip causing additional ripple on the output voltage. Based on previous testing, with a 2V difference between Vin and Vout, 20mA should be more than enough to keep switching and keep the boot capacitor charged.
Please see the following application note for more description and workarounds if it becomes an issue. http://www.ti.com/lit/an/slva547/slva547.pdf
Best Regards,
Anthony