#JavaScript (Node.js), 125 109 106 bytes#

-16 18 bytes from Zacharý

-1 by removing { and } by changing the incrementer to include the "set last to the current"

m=x=>{z=[[],[]];l=NaN;for(i=0;i<x.length;l=x[i++])if(l>x[i])z[1].push(x[i]);else z[0].push(x[i]);return z}

Basically, asks is the current item greater than the last item, add to the first list. Otherwise, add to the second.

Found out during this that comparing any number to NaN will always result false. Interesting!

Explanation:

m = x => {                         // Create function
  z = [[], []];                      // Initialize dropsort output
  l = NaN;                           // Initialize last element
  for (i = 0; i < x.length; l=x[i++])// For each item in input...
    if (l > x[i])                    // If current item is greater than previous
      z[1].push(x[i]);               // Then add it to the first part of output
    else                             // Elsewise
      z[0].push(x[i]);               // Add it to the nonordered part of the dropsort
                                     // Set last item to current item
  }                                  // Repeat
  return z                           // Return finished dropsort
}                                    // End function

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JavaScript (Node.js), 125 109 106 bytes

-16 18 bytes from Zacharý

-1 by removing { and } by changing the incrementer to include the "set last to the current"

m=x=>{z=[[],[]];l=NaN;for(i=0;i<x.length;l=x[i++])if(l>x[i])z[1].push(x[i]);else z[0].push(x[i]);return z}

Basically, asks is the current item greater than the last item, add to the first list. Otherwise, add to the second.

Found out during this that comparing any number to NaN will always result false. Interesting!

Explanation:

m = x => {                         // Create function
  z = [[], []];                      // Initialize dropsort output
  l = NaN;                           // Initialize last element
  for (i = 0; i < x.length; l=x[i++])// For each item in input...
    if (l > x[i])                    // If current item is greater than previous
      z[1].push(x[i]);               // Then add it to the first part of output
    else                             // Elsewise
      z[0].push(x[i]);               // Add it to the nonordered part of the dropsort
                                     // Set last item to current item
  }                                  // Repeat
  return z                           // Return finished dropsort
}                                    // End function

Try it online!

#JavaScript (Node.js), 125 109 108 bytes#

-16 bytes from Zacharý

-1 by removing { and } by changing the incrementer to include the "set last to the current"

m=(x)=>{z=[[],[]];l=NaN;for(i=0;i<x.length;l=x[i++])if(l>x[i])z[1].push(x[i]);else z[0].push(x[i]);return z}

Basically, asks is the current item greater than the last item, add to the first list. Otherwise, add to the second.

Found out during this that comparing any number to NaN will always result false. Interesting!

Explanation:

m = (x) => {                         // Create function
  z = [[], []];                      // Initialize dropsort output
  l = NaN;                           // Initialize last element
  for (i = 0; i < x.length; l=x[i++])// For each item in input...
    if (l > x[i])                    // If current item is greater than previous
      z[1].push(x[i]);               // Then add it to the first part of output
    else                             // Elsewise
      z[0].push(x[i]);               // Add it to the nonordered part of the dropsort
                                     // Set last item to current item
  }                                  // Repeat
  return z                           // Return finished dropsort
}                                    // End function

Try it online!

#JavaScript (Node.js), 125 109 106 bytes#

-16 18 bytes from Zacharý

-1 by removing { and } by changing the incrementer to include the "set last to the current"

m=x=>{z=[[],[]];l=NaN;for(i=0;i<x.length;l=x[i++])if(l>x[i])z[1].push(x[i]);else z[0].push(x[i]);return z}

Basically, asks is the current item greater than the last item, add to the first list. Otherwise, add to the second.

Found out during this that comparing any number to NaN will always result false. Interesting!

Explanation:

m = x => {                         // Create function
  z = [[], []];                      // Initialize dropsort output
  l = NaN;                           // Initialize last element
  for (i = 0; i < x.length; l=x[i++])// For each item in input...
    if (l > x[i])                    // If current item is greater than previous
      z[1].push(x[i]);               // Then add it to the first part of output
    else                             // Elsewise
      z[0].push(x[i]);               // Add it to the nonordered part of the dropsort
                                     // Set last item to current item
  }                                  // Repeat
  return z                           // Return finished dropsort
}                                    // End function

Try it online!

#JavaScript (Node.js), 125 109 bytes#

-16 bytes from Zacharý

(x)=>{z=[[],[]];l=NaN;for(i=0;i<x.length;i++){if(l>x[i])z[1].push(x[i]);else z[0].push(x[i]);l=x[i]}return z}

Basically, asks is the current item greater than the last item, add to the first list. Otherwise, add to the second.

Found out during this that comparing any number to NaN will always result false. Interesting!

Explanation:

(x) => {                           // Create function
  z = [[], []];                    // Initialize dropsort output
  l = NaN;                         // Initialize last element
  for (i = 0; i < x.length; i++) { // For each item in input...
    if (l > x[i])                  // If current item is greater than previous
      z[1].push(x[i]);             // Then add it to the first part of output
    else                           // Elsewise
      z[0].push(x[i]);             // Add it to the nonordered part of the dropsort
    l=x[i];                        // Set last item to current item
  }                                // Repeat
  return z                         // Return finished dropsort
}                                  // End function

Try it online!

#JavaScript (Node.js), 125 109 108 bytes#

-16 bytes from Zacharý

-1 by removing { and } by changing the incrementer to include the "set last to the current"

m=(x)=>{z=[[],[]];l=NaN;for(i=0;i<x.length;l=x[i++])if(l>x[i])z[1].push(x[i]);else z[0].push(x[i]);return z}

Basically, asks is the current item greater than the last item, add to the first list. Otherwise, add to the second.

Found out during this that comparing any number to NaN will always result false. Interesting!

Explanation:

m = (x) => {                         // Create function
  z = [[], []];                      // Initialize dropsort output
  l = NaN;                           // Initialize last element
  for (i = 0; i < x.length; l=x[i++])// For each item in input...
    if (l > x[i])                    // If current item is greater than previous
      z[1].push(x[i]);               // Then add it to the first part of output
    else                             // Elsewise
      z[0].push(x[i]);               // Add it to the nonordered part of the dropsort
                                     // Set last item to current item
  }                                  // Repeat
  return z                           // Return finished dropsort
}                                    // End function

Try it online!