Will software-based DSSS work?
I am not an expert in the field, but I don't think you can do DSSS in software (with this chip).  DSSS despreading requires the analog values of the received carrier as they add and subtract linearly with multiple stations transmitting simultaneously. If you can't get access to the analog level of the received carrier, you can't apply the math with the key to recover your data. Even if your GFSK receiver saturates at 1 or -1, you still need the mid-level 0 value on reception as part of the data to be used with your key to de-spread it. (As an example where it can be done, take a look at Software Defined Radios, these devices can sample the full-width spectrum and thus have the entire analog signal in a digital form, and can do DSSS; many GPS receivers are done this way) 

As for someone transmitting and jamming you, if they transmit a high power on a single channel that will not affect your spread-spectrum data stream, but if they transmit spread-spectrum at a higher power, it will just add or subtract with your own carrier and your data can be recovered. 

If they transmit (at any power level) and don't have your PN key, they will be essentially ignored by the despreader. I'll demonstrate this below.  But if the bad guy is using high enough spread-spectrum power to overload your receiver, then you will likely be stuck, and not much can be done about it anyways. 

Let's use a simple example.. we have a PN generator that creates the following bitstream: 001111000010101101110100001100 and we use 6 of these bits from the stream per 1 data bit. The 6 bits are called chips so as not to confuse them with data bits. So 1 bit is 6 chips and the chip rate is 6x the data rate. We'll use the bits like this: 001111 000010 101101 110100 001100 

To send the data sequence = {0,1,1,0,1}, it would be spread with an XOR of the chips from the PN generator: 

0 ⊕ 001111 = 110000  = [ 1  1 -1 -1 -1 -1 ]  ( bit 0 )
1 ⊕ 000010 = 000010  = [-1 -1 -1 -1  1 -1 ]  ( bit 1 )
1 ⊕ 101101 = 101101  = [ 1 -1  1  1 -1  1 ]  ( bit 2 )
0 ⊕ 110100 = 001011  = [-1 -1  1 -1  1  1 ]  ( bit 3 )
1 ⊕ 001100 = 001100  = [-1 -1  1  1 -1 -1 ]  ( bit 4 )


if the bad guy also transmits a spread sequence of 01010101.... then it will be sent like this: 

[ -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 ... ]


and no matter how it aligns with your sequence, your sequence can be recovered. Your carrier and the bad guy's carrier will add or subtract linearly, the data can be recovered after doing the inner product and normalizing the values. Continuing with your example:

0: [ 1  1 -1 -1 -1 -1 ]  ( your data bit 0 )
+: [ 1 -1  1 -1  1 -1 ]  ( the bad guy trying to overwrite you )
=: [ 2  0  0 -2  0 -2 ]  ( what your receiver sees )


If you take what your receiver sees, and apply the inner product with your key for bit 0, which is 001111 or [-1 -1  1  1  1  1] you get this: 

bit 0 = [ 2  0  0 -2  0 -2 ] * [-1 -1  1  1  1  1] = (-2+0+0-2+0-2) = -6 =&gt; -1 =&gt; 0


I'll do the second bit with 2 bad guys trying to overwrite you: 

1: [-1 -1 -1 -1  1 -1 ]  ( your data bit 1 )
+: [ 1 -1  1 -1  1 -1 ]  ( the bad guy trying to overwrite you )
+: [-1  1  1 -1 -1  1 ]  ( second bad guy ! ) 
=: [-1 -1  1 -3  1 -1 ]  ( what your receiver sees )


Again, take what your receiver sees and apply the inner product with the key for bit 1, which is 000010, or [-1 -1 -1 -1  1 -1]: 

bit 1 = [-1 -1  1 -3  1 -1 ] * [-1 -1 -1 -1  1 -1] = (1+1-1+3+1+1) = 6 =&gt; 1 =&gt; 1 


So bit \$0\$ is recovered as \$0\$, bit \$1\$ is recovered as \$1\$, even with the bad guys trying to overwrite the data stream. This is the essence of DSSS, and I'll leave the rest of the data as an exercise. It is quite immune to interference. However, for you to do this in software you can see that you need to have access to the analog values of the carrier level, like [-2,+3, 0 ...] and not just the bits. 

This chip only gives access to the demodulated GFSK bitstream, which is not enough information. As pointed out in the comments, it can be done in software if you have access to the full bandwidth data.