I don't think there is a problem. The individual files have this statement:
"* ARM Limited (ARM) is supplying this software for use with Cortex-M
* processor based microcontrollers. This file can be freely distributed
* within development tools that are supporting such ARM based processors."
and I don't see anything in the license that precludes it, either.
There is a gcc directory in the CMSIS distribution, which implies that it can be supplied with open-source tools.
(i) use and copy the CMSIS Specification for the purpose of developing, having developed, manufacturing,
having manufactured, offering to sell, selling, supplying or otherwise distributing products that comply with the
CMSIS Specification, provided that you preserve any copyright notices which are included with, or in, the CMSIS
Specification and provided that you do not use ARM's name, logo or trademarks to market such products;
implies that you can incorporate CMSIS in your open source project.