[android] How to implement an android:background that doesn't stretch?

Use draw9patch... included within Android Studio's SDK tools. You can define the stretchable areas of your image. Important parts are constrained and the image doesn't look all warped. A good demo on dra9patch is HERE

Use draw9patch to change your existing splash.png into new_splash.9.png, drag new_splash.9.png into the drawable-hdpi project folder ensure the AndroidManifest and styles.xml are proper as below:

AndroidManifest.xml:

<application
...
        android:theme="@style/splashScreenStyle"
>

styles.xml:

<style name="splashScreenStyle" parent="Theme.AppCompat.Light.DarkActionBar">
    <item name="android:windowBackground">@drawable/new_splash</item>
</style>