Does AES-ECB with random padding added to each block satisfy IND-CPA?
First, I would not call this AES-ECB, since adding randomness means that it is not ECB. Second, when limiting the message space to what fits into one block, asymptotically -- meaning for a pseudorandom permutation with block size $n$ -- this construction is actually even CCA-secure (and so non-malleable, since indistinguishability and non-malleability are equivalent under CCA attacks). This is actually Exercise 4.2.5 in Introduction to Modern Cryptography, 2nd edition. (When using multiple blocks as in the question, it would be CPA-secure and certainly malleable.)

I just want to note (with respect to @poncho's answer) that this is asymptotically secure. This means that when considering a pseudorandom permutation with block size $n$, no adversary running in time polynomial in $n$ can succeed with probability that is more than negligible in $n$. In practice, with a 128-bit block cipher, this construction will give very poor bounds. However, $2^{32}$ is really here $2^{n/4}$ which is super-polynomial. Thus, there is no contradiction.

In conclusion, this isn't a great idea in practice (unless you have a very large block cipher). However, it works in theory.