First I describe the situation as I understood it. You have measurements (not assumed to have a normal or any other distribution) on $n$ individuals, six observations on each individual, on two different conditions $A,B$.  We can write this as 
$$
   y_{ijA}, y_{ijB}
$$
for $i=1,2,\dotsc,n$, $j=1,2,3$.  This could be modelled as an ANOVA with one random and one fixed factor, we can write a linear model like
$$
  y_{ijk} = \mu + \eta_i + \beta I(\text{$k=A$}) +\epsilon_{ijk}
$$
This is one way of taking care of (that is, modeling) the dependence of the observations pertaining to the same individual. Here $\eta_i$ is a random effect for each individual and $\epsilon_{ijk}$ is the error term. (It might need some extra restrictions for identifiability).  This could be estimated with standard software for linear mixed models, like `lme4` in R.  I don't know about nonparametric tests for such models ... but you could use bootstrapping, maybe, or bayesian methods.  For references, look at any book about mixed models, if you are using R then maybe:  https://www.amazon.com/Mixed-Effects-Models-S-PLUS-Statistics-Computing/dp/1441903178/ref=sr_1_1?ie=UTF8&qid=1494104780&sr=8-1&keywords=bates+mixed+s-plus  it is accessible and very good.