As far as I know, Oracle does not, and never has, and probably never will in a near future provide an Instant Client for ARM-based Linux. As it is a proprietary software, there is little chances that you could "compile" it...
An option would be to write your own driver implementing of Oracle's wire-protocol. But this is far from trivial. Not mentioning the (possible) legal implications as this is a proprietary protocol.
For now, your best bet if you want to connect directly from your ARM box to an Oracle server, is probably to use the JDBC thin-driver as it is pure-java and should run on a JVM for ARM. If your application is not written in Java, you will probably need to write some kind of gateway yourself -- or wrap your own stuff through JNI maybe (sounds like a kludge, no?)
Depending on your needs and your project requirements, maybe you should investigate the option of having some kind of "web service" acting as a gateway to Oracle and running on an x86/amd64 box somewhere on your network. Then your clients (ARM-based or not) would access to the underlying DB through it.
There are many clients for open-sources RDMBS that you can use on ARM devices. Maybe you could manage to synchronize data between Oracle an one of these RDBMS ?
As of myself, I would push toward the third solution. But once again, this is all depending on your actual needs.
Given your various comments below, I would say that the choice of an ARM target was an error given your absolute need to embed instant client to connect to an Oracle RDBMS.
Maybe a small form-factor Intel's Atom based board would have been a better choice here? A Take a look for example at MinnowBoard or even Intel Galileao. Those are only suggestions. And I never worked with any of them. You will probably be able to find other/better options by googling a little. Check for the Linux compatibility/ease of install -- and you will have a full fledged x86 architecture at hand.
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