<?xml version="1.0" encoding="utf-8"?>
<!--Generated by DFIN XBRL Instance Document - http://www.dfinsolutions.com/ - Version R3.0 - on 12-June-2026 [04:57:05] {PM}-->
<schema targetNamespace="http://www.blackrock.com/20260612" elementFormDefault="qualified" xmlns:bpif="http://www.blackrock.com/20260612" xmlns="http://www.w3.org/2001/XMLSchema" xmlns:link="http://www.xbrl.org/2003/linkbase" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xbrli="http://www.xbrl.org/2003/instance" xmlns:xbrldt="http://xbrl.org/2005/xbrldt" xmlns:xl="http://www.xbrl.org/2003/XLink" xmlns:us-gaap="http://fasb.org/us-gaap/2025" xmlns:country="http://xbrl.sec.gov/country/2025" xmlns:currency="http://xbrl.sec.gov/currency/2025" xmlns:dei="http://xbrl.sec.gov/dei/2025" xmlns:exch="http://xbrl.sec.gov/exch/2025" xmlns:naics="http://xbrl.sec.gov/naics/2025" xmlns:sic="http://xbrl.sec.gov/sic/2025" xmlns:us-types="http://fasb.org/us-types/2025" xmlns:stpr="http://xbrl.sec.gov/stpr/2025" xmlns:deprecated="http://www.xbrl.org/2009/arcrole/fact-explanatoryFact" xmlns:srt="http://fasb.org/srt/2025" xmlns:srt-types="http://fasb.org/srt-types/2025" xmlns:dtr-types="http://www.xbrl.org/dtr/type/2024-01-31" xmlns:cef="http://xbrl.sec.gov/cef/2025" xmlns:cef-pre="http://xbrl.sec.gov/cef-pre/2025">
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        <link:usedOn>link:definitionLink</link:usedOn>
        <link:usedOn>link:calculationLink</link:usedOn>
      </link:roleType>
      <link:linkbase id="lnk" >
        <link:roleRef xlink:type="simple" xlink:href="bpif-20260612.xsd#Cover" roleURI="http://www.blackrock.com/role/Cover"  />
        <link:presentationLink xlink:type="extended" xlink:role="http://www.blackrock.com/role/Cover" >
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          <link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_CoverAbstract_2" xlink:to="loc_dei_CityAreaCode_16" use="optional" order="12" preferredLabel="http://www.xbrl.org/2003/role/label" />
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          <link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_CoverAbstract_2" xlink:to="loc_dei_SolicitingMaterial_19" use="optional" order="15" preferredLabel="http://www.xbrl.org/2003/role/label" />
          <link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_PreCommencementTenderOffer" xlink:label="loc_dei_PreCommencementTenderOffer_20" />
          <link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_CoverAbstract_2" xlink:to="loc_dei_PreCommencementTenderOffer_20" use="optional" order="16" preferredLabel="http://www.xbrl.org/2003/role/label" />
          <link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_PreCommencementIssuerTenderOffer" xlink:label="loc_dei_PreCommencementIssuerTenderOffer_21" />
          <link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_CoverAbstract_2" xlink:to="loc_dei_PreCommencementIssuerTenderOffer_21" use="optional" order="17" preferredLabel="http://www.xbrl.org/2003/role/label" />
          <link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityEmergingGrowthCompany" xlink:label="loc_dei_EntityEmergingGrowthCompany_25" />
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          <link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_AmendmentFlag" xlink:label="loc_dei_AmendmentFlag_26" />
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        <link:labelLink xlink:type="extended" xlink:role="http://www.xbrl.org/2003/role/link"  />
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  <import namespace="http://www.xbrl.org/2003/instance" schemaLocation="http://www.xbrl.org/2003/xbrl-instance-2003-12-31.xsd" />
  <import namespace="http://www.xbrl.org/2003/linkbase" schemaLocation="http://www.xbrl.org/2003/xbrl-linkbase-2003-12-31.xsd" />
  <import namespace="http://xbrl.org/2005/xbrldt" schemaLocation="http://www.xbrl.org/2005/xbrldt-2005.xsd" />
  <import namespace="http://fasb.org/us-gaap/2025" schemaLocation="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd" />
  <import namespace="http://xbrl.sec.gov/country/2025" schemaLocation="https://xbrl.sec.gov/country/2025/country-2025.xsd" />
  <import namespace="http://xbrl.sec.gov/currency/2025" schemaLocation="https://xbrl.sec.gov/currency/2025/currency-2025.xsd" />
  <import namespace="http://xbrl.sec.gov/exch/2025" schemaLocation="https://xbrl.sec.gov/exch/2025/exch-2025.xsd" />
  <import namespace="http://xbrl.sec.gov/naics/2025" schemaLocation="https://xbrl.sec.gov/naics/2025/naics-2025.xsd" />
  <import namespace="http://xbrl.sec.gov/sic/2025" schemaLocation="https://xbrl.sec.gov/sic/2025/sic-2025.xsd" />
  <import namespace="http://xbrl.sec.gov/stpr/2025" schemaLocation="https://xbrl.sec.gov/stpr/2025/stpr-2025.xsd" />
  <import namespace="http://www.xbrl.org/2009/role/negated" schemaLocation="http://www.xbrl.org/lrr/role/negated-2009-12-16.xsd" />
  <import namespace="http://www.xbrl.org/2009/role/net" schemaLocation="http://www.xbrl.org/lrr/role/net-2009-12-16.xsd" />
  <import namespace="http://www.xbrl.org/2009/arcrole/fact-explanatoryFact" schemaLocation="http://www.xbrl.org/lrr/arcrole/factExplanatory-2009-12-16.xsd" />
  <import namespace="http://xbrl.sec.gov/dei/2025" schemaLocation="https://xbrl.sec.gov/dei/2025/dei-2025.xsd" />
  <import namespace="http://fasb.org/us-types/2025" schemaLocation="https://xbrl.fasb.org/us-gaap/2025/elts/us-types-2025.xsd" />
  <import namespace="http://fasb.org/srt-types/2025" schemaLocation="https://xbrl.fasb.org/srt/2025/elts/srt-types-2025.xsd" />
  <import namespace="http://fasb.org/srt/2025" schemaLocation="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd" />
  <import namespace="http://www.xbrl.org/dtr/type/2024-01-31" schemaLocation="https://www.xbrl.org/dtr/type/2024-01-31/types.xsd" />
  <import namespace="http://xbrl.sec.gov/cef/2025" schemaLocation="https://xbrl.sec.gov/cef/2025/cef-2025.xsd" />
  <import namespace="http://xbrl.sec.gov/cef-pre/2025" schemaLocation="https://xbrl.sec.gov/cef/2025/cef-2025_pre.xsd" />
  <import namespace="http://xbrl.sec.gov/dei-lab/2025" schemaLocation="https://xbrl.sec.gov/dei/2025/dei-2025_lab.xsd" />
</schema>
