I happily booted Tiny Core 5.2 from a pen drive and then installed it on to the internal flash memory.
[Update May 2016] I had problems with running current distributions on this hardware. See the Netvoyager LX-1022 for an explanation.
[Update October 2017]. As expected the problem has now gone away with subsequent releases of the kernel. I find that Tiny Core 8.1.1 (using Linux kernel 4.8.17) runs with a CPU clock of 1GHz on the A3500 without any problems.